def generate_approx_poly_near_zero(self, function, high_bound, error_bound,
variable):
""" Generate polynomial approximation scheme """
error_function = lambda p, f, ai, mod, t: sollya.dirtyinfnorm(
p - f, ai)
# Some issues encountered when 0 is one of the interval bound
# so we use a symetric interval around it
approx_interval = Interval(2**-100, high_bound)
local_function = function / sollya.x
degree = sollya.sup(
sollya.guessdegree(local_function, approx_interval, error_bound))
degree_list = range(0, int(degree) + 4, 2)
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
function / sollya.x,
degree_list, [1] + [self.precision] * (len(degree_list) - 1),
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(
Log.Info, "approximation poly: {}\n with error {}".format(
poly_object, approx_error))
poly_scheme = Multiplication(
variable,
PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, variable, self.precision))
return poly_scheme, approx_error
def piecewise_approximation_degree_generator(function,
bound_low=-1.0,
bound_high=1.0,
num_intervals=16,
max_degree=2,
error_threshold=S2**-24):
""" """
interval_size = (bound_high - bound_low) / num_intervals
for i in range(num_intervals):
subint_low = bound_low + i * interval_size
subint_high = bound_low + (i + 1) * interval_size
local_function = function(sollya.x + subint_low)
local_interval = Interval(-interval_size, interval_size)
local_degree = sollya.guessdegree(local_function, local_interval,
error_threshold)
yield int(sollya.sup(local_degree))
def __init__(self,
precision = ML_Binary32,
abs_accuracy = S2**-24,
libm_compliant = True,
debug_flag = False,
fuse_fma = True,
fast_path_extract = True,
target = GenericProcessor(),
output_file = "expf.c",
function_name = "expf"):
# declaring target and instantiating optimization engine
processor = target
self.precision = precision
opt_eng = OptimizationEngine(processor)
gappacg = GappaCodeGenerator(processor, declare_cst = True, disable_debug = True)
# declaring CodeFunction and retrieving input variable
self.function_name = function_name
exp_implementation = CodeFunction(self.function_name, output_format = self.precision)
vx = exp_implementation.add_input_variable("x", self.precision)
Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
test_nan_or_inf = Test(vx, specifier = Test.IsInfOrNaN, likely = False, debug = True, tag = "nan_or_inf")
test_nan = Test(vx, specifier = Test.IsNaN, debug = True, tag = "is_nan_test")
test_positive = Comparison(vx, 0, specifier = Comparison.GreaterOrEqual, debug = True, tag = "inf_sign")
test_signaling_nan = Test(vx, specifier = Test.IsSignalingNaN, debug = True, tag = "is_signaling_nan")
return_snan = Statement(ExpRaiseReturn(ML_FPE_Invalid, return_value = FP_QNaN(self.precision)))
# return in case of infinity input
infty_return = Statement(ConditionBlock(test_positive, Return(FP_PlusInfty(self.precision)), Return(FP_PlusZero(self.precision))))
# return in case of specific value input (NaN or inf)
specific_return = ConditionBlock(test_nan, ConditionBlock(test_signaling_nan, return_snan, Return(FP_QNaN(self.precision))), infty_return)
# return in case of standard (non-special) input
# exclusion of early overflow and underflow cases
precision_emax = self.precision.get_emax()
precision_max_value = S2 * S2**precision_emax
exp_overflow_bound = ceil(log(precision_max_value))
early_overflow_test = Comparison(vx, exp_overflow_bound, likely = False, specifier = Comparison.Greater)
early_overflow_return = Statement(ClearException(), ExpRaiseReturn(ML_FPE_Inexact, ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)))
precision_emin = self.precision.get_emin_subnormal()
precision_min_value = S2 ** precision_emin
exp_underflow_bound = floor(log(precision_min_value))
early_underflow_test = Comparison(vx, exp_underflow_bound, likely = False, specifier = Comparison.Less)
early_underflow_return = Statement(ClearException(), ExpRaiseReturn(ML_FPE_Inexact, ML_FPE_Underflow, return_value = FP_PlusZero(self.precision)))
sollya_prec_map = {ML_Binary32: sollya.binary32, ML_Binary64: sollya.binary64}
# constant computation
invlog2 = round(1/log(2), sollya_prec_map[self.precision], RN)
interval_vx = Interval(exp_underflow_bound, exp_overflow_bound)
interval_fk = interval_vx * invlog2
interval_k = Interval(floor(inf(interval_fk)), ceil(sup(interval_fk)))
log2_hi_precision = self.precision.get_field_size() - (ceil(log2(sup(abs(interval_k)))) + 2)
Log.report(Log.Info, "log2_hi_precision: "), log2_hi_precision
invlog2_cst = Constant(invlog2, precision = self.precision)
log2_hi = round(log(2), log2_hi_precision, sollya.RN)
log2_lo = round(log(2) - log2_hi, sollya_prec_map[self.precision], sollya.RN)
# argument reduction
unround_k = vx * invlog2
unround_k.set_attributes(tag = "unround_k", debug = ML_Debug(display_format = "%f"))
k = NearestInteger(unround_k, precision = self.precision, debug = ML_Debug(display_format = "%f"))
ik = NearestInteger(unround_k, precision = ML_Int32, debug = ML_Debug(display_format = "%d"), tag = "ik")
ik.set_tag("ik")
k.set_tag("k")
exact_pre_mul = (k * log2_hi)
exact_pre_mul.set_attributes(exact= True)
exact_hi_part = vx - exact_pre_mul
exact_hi_part.set_attributes(exact = True)
r = exact_hi_part - k * log2_lo
r.set_tag("r")
r.set_attributes(debug = ML_Debug(display_format = "%f"))
opt_r = opt_eng.optimization_process(r, self.precision, copy = True, fuse_fma = fuse_fma)
tag_map = {}
opt_eng.register_nodes_by_tag(opt_r, tag_map)
cg_eval_error_copy_map = {
vx: Variable("x", precision = self.precision, interval = interval_vx),
tag_map["k"]: Variable("k", interval = interval_k, precision = self.precision)
}
#try:
if 1:
#eval_error = gappacg.get_eval_error(opt_r, cg_eval_error_copy_map, gappa_filename = "red_arg.g")
eval_error = gappacg.get_eval_error_v2(opt_eng, opt_r, cg_eval_error_copy_map, gappa_filename = "red_arg.g")
Log.report(Log.Info, "eval error: %s" % eval_error)
#except:
# Log.report(Log.Info, "gappa error evaluation failed")
print r.get_str(depth = None, display_precision = True, display_attribute = True)
print opt_r.get_str(depth = None, display_precision = True, display_attribute = True)
approx_interval = Interval(-log(2)/2, log(2)/2)
local_ulp = sup(ulp(exp(approx_interval), self.precision))
print "ulp: ", local_ulp
error_goal = local_ulp #S2**-(self.precision.get_field_size()+1)
error_goal_approx = S2**-1 * error_goal
Log.report(Log.Info, "\033[33;1m building mathematical polynomial \033[0m\n")
poly_degree = sup(guessdegree(exp(x), approx_interval, error_goal_approx)) #- 1
init_poly_degree = poly_degree
return
while 1:
Log.report(Log.Info, "attempting poly degree: %d" % poly_degree)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(exp(x), poly_degree, [self.precision]*(poly_degree+1), approx_interval, absolute)
Log.report(Log.Info, "poly approx error: %s" % poly_approx_error)
Log.report(Log.Info, "\033[33;1m generating polynomial evaluation scheme \033[0m")
poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, r, unified_precision = self.precision)
poly.set_tag("poly")
# optimizing poly before evaluation error computation
opt_poly = opt_eng.optimization_process(poly, self.precision)
#print "poly: ", poly.get_str(depth = None, display_precision = True)
#print "opt_poly: ", opt_poly.get_str(depth = None, display_precision = True)
# evaluating error of the polynomial approximation
r_gappa_var = Variable("r", precision = self.precision, interval = approx_interval)
poly_error_copy_map = {
r.get_handle().get_node(): r_gappa_var
}
gappacg = GappaCodeGenerator(target, declare_cst = False, disable_debug = True)
poly_eval_error = gappacg.get_eval_error_v2(opt_eng, poly.get_handle().get_node(), poly_error_copy_map, gappa_filename = "gappa_poly.g")
Log.report(Log.Info, "poly evaluation error: %s" % poly_eval_error)
global_poly_error = poly_eval_error + poly_approx_error
global_rel_poly_error = global_poly_error / exp(approx_interval)
print "global_poly_error: ", global_poly_error, global_rel_poly_error
flag = local_ulp > sup(abs(global_rel_poly_error))
print "test: ", flag
if flag: break
else:
if poly_degree > init_poly_degree + 5:
Log.report(Log.Error, "poly degree search did not converge")
poly_degree += 1
late_overflow_test = Comparison(ik, self.precision.get_emax(), specifier = Comparison.Greater, likely = False, debug = True, tag = "late_overflow_test")
overflow_exp_offset = (self.precision.get_emax() - self.precision.get_field_size() / 2)
diff_k = ik - overflow_exp_offset
diff_k.set_attributes(debug = ML_Debug(display_format = "%d"), tag = "diff_k")
late_overflow_result = (ExponentInsertion(diff_k) * poly) * ExponentInsertion(overflow_exp_offset)
late_overflow_result.set_attributes(silent = False, tag = "late_overflow_result", debug = debugf)
late_overflow_return = ConditionBlock(Test(late_overflow_result, specifier = Test.IsInfty, likely = False), ExpRaiseReturn(ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)), Return(late_overflow_result))
late_underflow_test = Comparison(k, self.precision.get_emin_normal(), specifier = Comparison.LessOrEqual, likely = False)
underflow_exp_offset = 2 * self.precision.get_field_size()
late_underflow_result = (ExponentInsertion(ik + underflow_exp_offset) * poly) * ExponentInsertion(-underflow_exp_offset)
late_underflow_result.set_attributes(debug = ML_Debug(display_format = "%e"), tag = "late_underflow_result", silent = False)
test_subnormal = Test(late_underflow_result, specifier = Test.IsSubnormal)
late_underflow_return = Statement(ConditionBlock(test_subnormal, ExpRaiseReturn(ML_FPE_Underflow, return_value = late_underflow_result)), Return(late_underflow_result))
std_result = poly * ExponentInsertion(ik, tag = "exp_ik", debug = debug_lftolx)
std_result.set_attributes(tag = "std_result", debug = debug_lftolx)
result_scheme = ConditionBlock(late_overflow_test, late_overflow_return, ConditionBlock(late_underflow_test, late_underflow_return, Return(std_result)))
std_return = ConditionBlock(early_overflow_test, early_overflow_return, ConditionBlock(early_underflow_test, early_underflow_return, result_scheme))
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = ConditionBlock(test_nan_or_inf, Statement(ClearException(), specific_return), std_return)
#print scheme.get_str(depth = None, display_precision = True)
# fusing FMA
if fuse_fma:
Log.report(Log.Info, "\033[33;1m MDL fusing FMA \033[0m")
scheme = opt_eng.fuse_multiply_add(scheme, silence = True)
Log.report(Log.Info, "\033[33;1m MDL abstract scheme \033[0m")
opt_eng.instantiate_abstract_precision(scheme, None)
Log.report(Log.Info, "\033[33;1m MDL instantiated scheme \033[0m")
opt_eng.instantiate_precision(scheme, default_precision = self.precision)
Log.report(Log.Info, "\033[33;1m subexpression sharing \033[0m")
opt_eng.subexpression_sharing(scheme)
Log.report(Log.Info, "\033[33;1m silencing operation \033[0m")
opt_eng.silence_fp_operations(scheme)
# registering scheme as function implementation
exp_implementation.set_scheme(scheme)
# check processor support
Log.report(Log.Info, "\033[33;1m checking processor support \033[0m")
opt_eng.check_processor_support(scheme)
# factorizing fast path
if fast_path_extract:
Log.report(Log.Info, "\033[33;1m factorizing fast path\033[0m")
opt_eng.factorize_fast_path(scheme)
Log.report(Log.Info, "\033[33;1m generating source code \033[0m")
cg = CCodeGenerator(processor, declare_cst = False, disable_debug = not debug_flag, libm_compliant = libm_compliant)
self.result = exp_implementation.get_definition(cg, C_Code, static_cst = True)
#self.result.add_header("support_lib/ml_types.h")
self.result.add_header("support_lib/ml_special_values.h")
self.result.add_header_comment("polynomial degree for exp(x): %d" % poly_degree)
self.result.add_header_comment("sollya polynomial for exp(x): %s" % poly_object.get_sollya_object())
if debug_flag:
self.result.add_header("stdio.h")
self.result.add_header("inttypes.h")
output_stream = open(output_file, "w")#"%s.c" % exp_implementation.get_name(), "w")
output_stream.write(self.result.get(cg))
output_stream.close()
def generate_scheme(self):
#func_implementation = CodeFunction(self.function_name, output_format = self.precision)
vx = self.implementation.add_input_variable("x", self.get_input_precision())
sollya_precision = self.get_sollya_precision()
# retrieving processor inverse approximation table
#dummy_var = Variable("dummy", precision = self.precision)
#dummy_div_seed = DivisionSeed(dummy_var, precision = self.precision)
#inv_approx_table = self.processor.get_recursive_implementation(dummy_div_seed, language = None, table_getter = lambda self: self.approx_table_map)
lo_bound_global = SollyaObject(0.0)
hi_bound_global = SollyaObject(0.75)
approx_interval = Interval(lo_bound_global, hi_bound_global)
approx_interval_size = hi_bound_global - lo_bound_global
# table creation
table_index_size = 7
field_index_size = 2
exp_index_size = table_index_size - field_index_size
table_size = 2**table_index_size
table_index_range = range(table_size)
local_degree = 9
coeff_table = ML_Table(dimensions = [table_size, local_degree], storage_precision = self.precision)
#local_interval_size = approx_interval_size / SollyaObject(table_size)
#for i in table_index_range:
# degree = 6
# lo_bound = lo_bound_global + i * local_interval_size
# hi_bound = lo_bound_global + (i+1) * local_interval_size
# approx_interval = Interval(lo_bound, hi_bound)
# local_poly_object, local_error = Polynomial.build_from_approximation_with_error(acos(x), degree, [self.precision] * (degree+1), approx_interval, absolute)
# local_error = int(log2(sup(abs(local_error / acos(approx_interval)))))
# print approx_interval, local_error
exp_lo = 2**exp_index_size
for i in table_index_range:
lo_bound = (1.0 + (i % 2**field_index_size) * S2**-field_index_size) * S2**(i / 2**field_index_size - exp_lo)
hi_bound = (1.0 + ((i % 2**field_index_size) + 1) * S2**-field_index_size) * S2**(i / 2**field_index_size - exp_lo)
local_approx_interval = Interval(lo_bound, hi_bound)
local_poly_object, local_error = Polynomial.build_from_approximation_with_error(acos(1 - x), local_degree, [self.precision] * (local_degree+1), local_approx_interval, sollya.absolute)
local_error = int(log2(sup(abs(local_error / acos(1 - local_approx_interval)))))
coeff_table
print local_approx_interval, local_error
for d in xrange(local_degree):
coeff_table[i][d] = sollya.coeff(local_poly_object.get_sollya_object(), d)
table_index = BitLogicRightShift(vx, vx.get_precision().get_field_size() - field_index_size) - (exp_lo << field_index_size)
print "building mathematical polynomial"
poly_degree = sup(sollya.guessdegree(acos(x), approx_interval, S2**-(self.precision.get_field_size())))
print "guessed polynomial degree: ", int(poly_degree)
#global_poly_object = Polynomial.build_from_approximation(log10(1+x)/x, poly_degree, [self.precision]*(poly_degree+1), approx_interval, absolute)
print "generating polynomial evaluation scheme"
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
# building eval error map
#eval_error_map = {
# red_vx: Variable("red_vx", precision = self.precision, interval = red_vx.get_interval()),
# log_inv_hi: Variable("log_inv_hi", precision = self.precision, interval = table_high_interval),
# log_inv_lo: Variable("log_inv_lo", precision = self.precision, interval = table_low_interval),
#}
# computing gappa error
#poly_eval_error = self.get_eval_error(result, eval_error_map)
# main scheme
print "MDL scheme"
scheme = Statement(Return(vx))
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
sollya_precision = self.get_input_precision().get_sollya_object()
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# testing special value inputs
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
# if input is a signaling NaN, raise an invalid exception and returns
# a quiet NaN
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = self.precision.get_integer_format()
# log2(vx)
# r = vx_mant
# e = vx_exp
# vx reduced to r in [1, 2[
# log2(vx) = log2(r * 2^e)
# = log2(r) + e
#
## log2(r) is approximated by
# log2(r) = log2(inv_seed(r) * r / inv_seed(r)
# = log2(inv_seed(r) * r) - log2(inv_seed(r))
# inv_seed(r) in ]1/2, 1] => log2(inv_seed(r)) in ]-1, 0]
#
# inv_seed(r) * r ~ 1
# we can easily tabulate -log2(inv_seed(r))
#
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("inv_table"))
# value for index 0 is set to 0.0
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in range(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
#inv_value = (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
#print inv_approx_table[i][0], inv_value
inv_value = inv_approx_table[i][0]
value_high_bitsize = self.precision.get_field_size() - (
self.precision.get_exponent_size() + 1)
value_high = round(log2(inv_value), value_high_bitsize, sollya.RN)
value_low = round(
log2(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_lftolx)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debugd)
# The main table is indexed by the 7 most significant bits
# of the mantissa
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debuglld)
# argument reduction
# Using AND -2 to exclude LSB set to 1 for Newton-Raphson convergence
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(DivisionSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_lftolx,
silent=True),
precision=ML_UInt64),
Constant(-2, precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
_red_vx = FMA(arg_red_index, _vx_mant, -1.0)
_red_vx.set_attributes(tag="_red_vx", debug=debug_lftolx)
inv_err = S2**-inv_approx_table.index_size
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log2(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() * 1.1))) + 1
sollya.settings.display = sollya.hexadecimal
global_poly_object, approx_error = Polynomial.build_from_approximation_with_error(
log2(1 + sollya.x) / sollya.x,
poly_degree, [self.precision] * (poly_degree + 1),
approx_interval,
sollya.absolute,
error_function=lambda p, f, ai, mod, t: sollya.dirtyinfnorm(
p - f, ai))
Log.report(
Log.Info, "poly_degree={}, approx_error={}".format(
poly_degree, approx_error))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
#poly_object = global_poly_object.sub_poly(start_index=0,offset=0)
Attributes.set_default_silent(True)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly = FMA(pre_poly, _red_vx,
global_poly_object.get_cst_coeff(0, self.precision))
_poly.set_attributes(tag="poly", debug=debug_lftolx)
Log.report(
Log.Verbose, "sollya global_poly_object: {}".format(
global_poly_object.get_sollya_object()))
Log.report(
Log.Verbose, "sollya poly_object: {}".format(
poly_object.get_sollya_object()))
corr_exp = _vx_exp if exp_corr_factor == None else _vx_exp + exp_corr_factor
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
pre_result = -_log_inv_hi + (_red_vx * _poly + (-_log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = Conversion(corr_exp, precision=self.precision)
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_hex",
debug=debug_lftolx)
_result = exact_log2_hi_exp + pre_result
return _result, _poly, _log_inv_lo, _log_inv_hi, _red_vx
result, poly, log_inv_lo, log_inv_hi, red_vx = compute_log(vx)
result.set_attributes(tag="result", debug=debug_lftolx)
# specific input value predicate
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="vx_snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="vx_inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debugd,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debugd,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debugd)
# Specific specific for the case exp == -1
# log2(x) = log2(m) - 1
#
# as m in [1, 2[, log2(m) in [0, 1[
# if r is close to 2, a catastrophic cancellation can occur
#
# r = seed(m)
# log2(x) = log2(seed(m) * m / seed(m)) - 1
# = log2(seed(m) * m) - log2(seed(m)) - 1
#
# for m really close to 2 => seed(m) = 0.5
# => log2(x) = log2(0.5 * m)
# =
result_exp_m1 = (-log_inv_hi - 1.0) + FMA(poly, red_vx, -log_inv_lo)
result_exp_m1.set_attributes(tag="result_exp_m1", debug=debug_lftolx)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
result_subnormal.set_attributes(tag="result_subnormal",
debug=debug_lftolx)
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log(1 + x) / x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_lftolx)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one", debug=debugd, likely=False)
# main scheme
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
),
Statement(ClearException(), result_subnormal,
Return(result_subnormal))),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result_exp_m1),
Return(result))))))
scheme = Statement(result, pre_scheme)
return scheme
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
Log.set_dump_stdout(True)
Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
C_m1 = Constant(-1, precision = self.precision)
test_NaN_or_inf = Test(vx, specifier = Test.IsInfOrNaN, likely = False, debug = debug_multi, tag = "NaN_or_inf", precision = ML_Bool)
test_NaN = Test(vx, specifier = Test.IsNaN, likely = False, debug = debug_multi, tag = "is_NaN", precision = ML_Bool)
test_inf = Comparison(vx, 0, specifier = Comparison.Greater, debug = debug_multi, tag = "sign", precision = ML_Bool, likely = False);
# Infnty input
infty_return = Statement(ConditionBlock(test_inf, Return(FP_PlusInfty(self.precision)), Return(C_m1)))
# non-std input (inf/nan)
specific_return = ConditionBlock(test_NaN, Return(FP_QNaN(self.precision)), infty_return)
# Over/Underflow Tests
precision_emax = self.precision.get_emax()
precision_max_value = S2**(precision_emax + 1)
expm1_overflow_bound = ceil(log(precision_max_value + 1))
overflow_test = Comparison(vx, expm1_overflow_bound, likely = False, specifier = Comparison.Greater, precision = ML_Bool)
overflow_return = Statement(Return(FP_PlusInfty(self.precision)))
precision_emin = self.precision.get_emin_subnormal()
precision_min_value = S2** precision_emin
expm1_underflow_bound = floor(log(precision_min_value) + 1)
underflow_test = Comparison(vx, expm1_underflow_bound, likely = False, specifier = Comparison.Less, precision = ML_Bool)
underflow_return = Statement(Return(C_m1))
sollya_precision = {ML_Binary32: sollya.binary32, ML_Binary64: sollya.binary64}[self.precision]
int_precision = {ML_Binary32: ML_Int32, ML_Binary64: ML_Int64}[self.precision]
# Constants
log_2 = round(log(2), sollya_precision, sollya.RN)
invlog2 = round(1/log(2), sollya_precision, sollya.RN)
log_2_cst = Constant(log_2, precision = self.precision)
interval_vx = Interval(expm1_underflow_bound, expm1_overflow_bound)
interval_fk = interval_vx * invlog2
interval_k = Interval(floor(inf(interval_fk)), ceil(sup(interval_fk)))
log2_hi_precision = self.precision.get_field_size() - 6
log2_hi = round(log(2), log2_hi_precision, sollya.RN)
log2_lo = round(log(2) - log2_hi, sollya_precision, sollya.RN)
# Reduction
unround_k = vx * invlog2
ik = NearestInteger(unround_k, precision = int_precision, debug = debug_multi, tag = "ik")
k = Conversion(ik, precision = self.precision, tag = "k")
red_coeff1 = Multiplication(k, log2_hi, precision = self.precision)
red_coeff2 = Multiplication(Negation(k, precision = self.precision), log2_lo, precision = self.precision)
pre_sub_mul = Subtraction(vx, red_coeff1, precision = self.precision)
s = Addition(pre_sub_mul, red_coeff2, precision = self.precision)
z = Subtraction(s, pre_sub_mul, precision = self.precision)
t = Subtraction(red_coeff2, z, precision = self.precision)
r = Addition(s, t, precision = self.precision)
r.set_attributes(tag = "r", debug = debug_multi)
r_interval = Interval(-log_2/S2, log_2/S2)
local_ulp = sup(ulp(exp(r_interval), self.precision))
print("ulp: ", local_ulp)
error_goal = S2**-1*local_ulp
print("error goal: ", error_goal)
# Polynomial Approx
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
Log.report(Log.Info, "\033[33;1m Building polynomial \033[0m\n")
poly_degree = sup(guessdegree(expm1(sollya.x), r_interval, error_goal) + 1)
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_degree_list = range(0, poly_degree)
precision_list = [self.precision] *(len(poly_degree_list) + 1)
poly_object, poly_error = Polynomial.build_from_approximation_with_error(expm1(sollya.x), poly_degree, precision_list, r_interval, sollya.absolute, error_function = error_function)
sub_poly = poly_object.sub_poly(start_index = 2)
Log.report(Log.Info, "Poly : %s" % sub_poly)
Log.report(Log.Info, "poly error : {} / {:d}".format(poly_error, int(sollya.log2(poly_error))))
pre_sub_poly = polynomial_scheme_builder(sub_poly, r, unified_precision = self.precision)
poly = r + pre_sub_poly
poly.set_attributes(tag = "poly", debug = debug_multi)
exp_k = ExponentInsertion(ik, tag = "exp_k", debug = debug_multi, precision = self.precision)
exp_mk = ExponentInsertion(-ik, tag = "exp_mk", debug = debug_multi, precision = self.precision)
diff = 1 - exp_mk
diff.set_attributes(tag = "diff", debug = debug_multi)
# Late Tests
late_overflow_test = Comparison(ik, self.precision.get_emax(), specifier = Comparison.Greater, likely = False, debug = debug_multi, tag = "late_overflow_test")
overflow_exp_offset = (self.precision.get_emax() - self.precision.get_field_size() / 2)
diff_k = ik - overflow_exp_offset
exp_diff_k = ExponentInsertion(diff_k, precision = self.precision, tag = "exp_diff_k", debug = debug_multi)
exp_oflow_offset = ExponentInsertion(overflow_exp_offset, precision = self.precision, tag = "exp_offset", debug = debug_multi)
late_overflow_result = (exp_diff_k * (1 + poly)) * exp_oflow_offset - 1.0
late_overflow_return = ConditionBlock(
Test(late_overflow_result, specifier = Test.IsInfty, likely = False),
ExpRaiseReturn(ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)),
Return(late_overflow_result)
)
late_underflow_test = Comparison(k, self.precision.get_emin_normal(), specifier = Comparison.LessOrEqual, likely = False)
underflow_exp_offset = 2 * self.precision.get_field_size()
corrected_coeff = ik + underflow_exp_offset
exp_corrected = ExponentInsertion(corrected_coeff, precision = self.precision)
exp_uflow_offset = ExponentInsertion(-underflow_exp_offset, precision = self.precision)
late_underflow_result = ( exp_corrected * (1 + poly)) * exp_uflow_offset - 1.0
test_subnormal = Test(late_underflow_result, specifier = Test.IsSubnormal, likely = False)
late_underflow_return = Statement(
ConditionBlock(
test_subnormal,
ExpRaiseReturn(ML_FPE_Underflow, return_value = late_underflow_result)),
Return(late_underflow_result)
)
# Reconstruction
std_result = exp_k * ( poly + diff )
std_result.set_attributes(tag = "result", debug = debug_multi)
result_scheme = ConditionBlock(
late_overflow_test,
late_overflow_return,
ConditionBlock(
late_underflow_test,
late_underflow_return,
Return(std_result)
)
)
std_return = ConditionBlock(
overflow_test,
overflow_return,
ConditionBlock(
underflow_test,
underflow_return,
result_scheme)
)
scheme = ConditionBlock(
test_NaN_or_inf,
Statement(specific_return),
std_return
)
return scheme
def generic_poly_split(offset_fct, indexing, target_eps, coeff_precision, vx):
""" generate the meta approximation for @p offset_fct over several
intervals defined by @p indexing object
For each sub-interval, a polynomial approximation with
maximal_error @p target_eps is tabulated, and evaluated using format
@p coeff_precision.
The input variable is @p vx """
# computing degree for a different polynomial approximation on each
# sub-interval
poly_degree_list = [
int(sup(guessdegree(offset_fct(offset), sub_interval, target_eps)))
for offset, sub_interval in indexing.get_offseted_sub_list()
]
poly_max_degree = max(poly_degree_list)
# tabulating polynomial coefficients on split_num sub-interval of interval
poly_table = ML_NewTable(
dimensions=[indexing.split_num, poly_max_degree + 1],
storage_precision=coeff_precision,
const=True)
offset_table = ML_NewTable(dimensions=[indexing.split_num],
storage_precision=coeff_precision,
const=True)
max_error = 0.0
for sub_index in range(indexing.split_num):
poly_degree = poly_degree_list[sub_index]
offset, approx_interval = indexing.get_offseted_sub_interval(sub_index)
offset_table[sub_index] = offset
if poly_degree == 0:
# managing constant approximation separately since it seems
# to break sollya
local_approx = coeff_precision.round_sollya_object(
offset_fct(offset)(inf(approx_interval)))
poly_table[sub_index][0] = local_approx
for monomial_index in range(1, poly_max_degree + 1):
poly_table[sub_index][monomial_index] = 0
approx_error = sollya.infnorm(
offset_fct(offset) - local_approx, approx_interval)
else:
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
offset_fct(offset), poly_degree,
[coeff_precision] * (poly_degree + 1), approx_interval,
sollya.relative)
for monomial_index in range(poly_max_degree + 1):
if monomial_index <= poly_degree:
poly_table[sub_index][
monomial_index] = poly_object.coeff_map[monomial_index]
else:
poly_table[sub_index][monomial_index] = 0
max_error = max(approx_error, max_error)
Log.report(Log.Debug, "max approx error is {}", max_error)
# indexing function: derive index from input @p vx value
poly_index = indexing.get_index_node(vx)
poly_index.set_attributes(tag="poly_index", debug=debug_multi)
ext_precision = get_extended_fp_precision(coeff_precision)
# building polynomial evaluation scheme
offset = TableLoad(offset_table,
poly_index,
precision=coeff_precision,
tag="offset",
debug=debug_multi)
poly = TableLoad(poly_table,
poly_index,
poly_max_degree,
precision=coeff_precision,
tag="poly_init",
debug=debug_multi)
red_vx = Subtraction(vx,
offset,
precision=vx.precision,
tag="red_vx",
debug=debug_multi)
for monomial_index in range(poly_max_degree, -1, -1):
coeff = TableLoad(poly_table,
poly_index,
monomial_index,
precision=coeff_precision,
tag="poly_%d" % monomial_index,
debug=debug_multi)
#fma_precision = coeff_precision if monomial_index > 1 else ext_precision
fma_precision = coeff_precision
poly = FMA(red_vx, poly, coeff, precision=fma_precision)
#return Conversion(poly, precision=coeff_precision)
#return poly.hi
return poly
def __init__(self,
precision=ML_Binary32,
abs_accuracy=S2**-24,
libm_compliant=True,
debug_flag=False,
fuse_fma=True,
fast_path_extract=True,
target=GenericProcessor(),
output_file="sinf.c",
function_name="sinf"):
# declaring CodeFunction and retrieving input variable
self.function_name = function_name
self.precision = precision
self.processor = target
func_implementation = CodeFunction(self.function_name,
output_format=self.precision)
vx = func_implementation.add_input_variable("x", self.precision)
sollya_precision = self.precision.sollya_object
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid, return_value=FP_QNaN(ML_Binary32)))
int_precision = ML_Int64 if self.precision is ML_Binary64 else ML_Int32
inv_pi_value = 1 / pi
# argument reduction
mod_pi_x = NearestInteger(vx * inv_pi_value)
red_vx = vx - mod_pi_x * pi
approx_interval = Interval(0, pi / 2)
poly_degree = sup(
guessdegree(
sin(sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
sin(sollya.x) / sollya.x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object #.sub_poly(start_index = 1)
print "generating polynomial evaluation scheme"
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_lftolx)
print global_poly_object.get_sollya_object()
pre_result = vx * _poly
result = pre_result
result.set_attributes(tag="result", debug=debug_lftolx)
# main scheme
print "MDL scheme"
scheme = Statement(Return(result))
#print scheme.get_str(depth = None, display_precision = True)
opt_eng = OptimizationEngine(self.processor)
# fusing FMA
print "MDL fusing FMA"
scheme = opt_eng.fuse_multiply_add(scheme, silence=True)
print "MDL abstract scheme"
opt_eng.instantiate_abstract_precision(scheme, None)
#print scheme.get_str(depth = None, display_precision = True)
print "MDL instantiated scheme"
opt_eng.instantiate_precision(scheme, default_precision=ML_Binary32)
print "subexpression sharing"
opt_eng.subexpression_sharing(scheme)
print "silencing operation"
opt_eng.silence_fp_operations(scheme)
# registering scheme as function implementation
func_implementation.set_scheme(scheme)
# check processor support
opt_eng.check_processor_support(scheme)
# factorizing fast path
opt_eng.factorize_fast_path(scheme)
#print scheme.get_str(depth = None, display_precision = True)
cg = CCodeGenerator(self.processor,
declare_cst=False,
disable_debug=not debug_flag,
libm_compliant=libm_compliant)
self.result = func_implementation.get_definition(cg,
C_Code,
static_cst=True)
#print self.result.get(cg)
output_stream = open("%s.c" % func_implementation.get_name(), "w")
output_stream.write(self.result.get(cg))
output_stream.close()
def generate_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = self.implementation.add_input_variable("x", self.precision)
Log.report(Log.Info, "generating implementation scheme")
if self.debug_flag:
Log.report(Log.Info, "debug has been enabled")
# local overloading of RaiseReturn operation
def SincosRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
sollya_precision = self.precision.get_sollya_object()
hi_precision = self.precision.get_field_size() - 8
cw_hi_precision = self.precision.get_field_size() - 4
ext_precision = {
ML_Binary32: ML_Binary64,
ML_Binary64: ML_Binary64
}[self.precision]
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
if self.precision is ML_Binary32:
ph_bound = S2**10
else:
ph_bound = S2**33
test_ph_bound = Comparison(vx,
ph_bound,
specifier=Comparison.GreaterOrEqual,
precision=ML_Bool,
likely=False)
# argument reduction
# m
frac_pi_index = {ML_Binary32: 10, ML_Binary64: 14}[self.precision]
C0 = Constant(0, precision=int_precision)
C1 = Constant(1, precision=int_precision)
C_offset = Constant(3 * S2**(frac_pi_index - 1),
precision=int_precision)
# 2^m / pi
frac_pi = round(S2**frac_pi_index / pi, cw_hi_precision, sollya.RN)
frac_pi_lo = round(S2**frac_pi_index / pi - frac_pi, sollya_precision,
sollya.RN)
# pi / 2^m, high part
inv_frac_pi = round(pi / S2**frac_pi_index, cw_hi_precision, sollya.RN)
# pi / 2^m, low part
inv_frac_pi_lo = round(pi / S2**frac_pi_index - inv_frac_pi,
sollya_precision, sollya.RN)
# computing k
vx.set_attributes(tag="vx", debug=debug_multi)
vx_pi = Addition(Multiplication(vx,
Constant(frac_pi,
precision=self.precision),
precision=self.precision),
Multiplication(vx,
Constant(frac_pi_lo,
precision=self.precision),
precision=self.precision),
precision=self.precision,
tag="vx_pi",
debug=debug_multi)
k = NearestInteger(vx_pi,
precision=int_precision,
tag="k",
debug=debug_multi)
# k in floating-point precision
fk = Conversion(k,
precision=self.precision,
tag="fk",
debug=debug_multi)
inv_frac_pi_cst = Constant(inv_frac_pi,
tag="inv_frac_pi",
precision=self.precision,
debug=debug_multi)
inv_frac_pi_lo_cst = Constant(inv_frac_pi_lo,
tag="inv_frac_pi_lo",
precision=self.precision,
debug=debug_multi)
# Cody-Waite reduction
red_coeff1 = Multiplication(fk,
inv_frac_pi_cst,
precision=self.precision,
exact=True)
red_coeff2 = Multiplication(Negation(fk, precision=self.precision),
inv_frac_pi_lo_cst,
precision=self.precision,
exact=True)
# Should be exact / Sterbenz' Lemma
pre_sub_mul = Subtraction(vx,
red_coeff1,
precision=self.precision,
exact=True)
# Fast2Sum
s = Addition(pre_sub_mul,
red_coeff2,
precision=self.precision,
unbreakable=True,
tag="s",
debug=debug_multi)
z = Subtraction(s,
pre_sub_mul,
precision=self.precision,
unbreakable=True,
tag="z",
debug=debug_multi)
t = Subtraction(red_coeff2,
z,
precision=self.precision,
unbreakable=True,
tag="t",
debug=debug_multi)
red_vx_std = Addition(s, t, precision=self.precision)
red_vx_std.set_attributes(tag="red_vx_std", debug=debug_multi)
# To compute sine we offset x by 3pi/2
# which means add 3 * S2^(frac_pi_index-1) to k
if self.sin_output:
Log.report(Log.Info, "Computing Sin")
offset_k = Addition(k,
C_offset,
precision=int_precision,
tag="offset_k")
else:
Log.report(Log.Info, "Computing Cos")
offset_k = k
modk = Variable("modk",
precision=int_precision,
var_type=Variable.Local)
red_vx = Variable("red_vx",
precision=self.precision,
var_type=Variable.Local)
# Faster modulo using bitwise logic
modk_std = BitLogicAnd(offset_k,
2**(frac_pi_index + 1) - 1,
precision=int_precision,
tag="modk",
debug=debug_multi)
approx_interval = Interval(-pi / (S2**(frac_pi_index + 1)),
pi / S2**(frac_pi_index + 1))
red_vx.set_interval(approx_interval)
Log.report(Log.Info, "approx interval: %s\n" % approx_interval)
Log.report(Log.Info,
"building tabulated approximation for sin and cos")
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
# polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
table_index_size = frac_pi_index + 1
cos_table = ML_NewTable(dimensions=[2**table_index_size, 1],
storage_precision=self.precision,
tag=self.uniquify_name("cos_table"))
for i in range(2**(frac_pi_index + 1)):
local_x = i * pi / S2**frac_pi_index
cos_local = round(cos(local_x), self.precision.get_sollya_object(),
sollya.RN)
cos_table[i][0] = cos_local
sin_index = Modulo(modk + 2**(frac_pi_index - 1),
2**(frac_pi_index + 1),
precision=int_precision,
tag="sin_index") #, debug = debug_multi)
tabulated_cos = TableLoad(cos_table,
modk,
C0,
precision=self.precision,
tag="tab_cos",
debug=debug_multi)
tabulated_sin = -TableLoad(cos_table,
sin_index,
C0,
precision=self.precision,
tag="tab_sin",
debug=debug_multi)
poly_degree_cos = sup(
guessdegree(cos(sollya.x), approx_interval, S2**
-self.precision.get_precision()) + 2)
poly_degree_sin = sup(
guessdegree(
sin(sollya.x) / sollya.x, approx_interval, S2**
-self.precision.get_precision()) + 2)
poly_degree_cos_list = range(0, int(poly_degree_cos) + 3)
poly_degree_sin_list = range(0, int(poly_degree_sin) + 3)
# cosine polynomial: limiting first and second coefficient precision to 1-bit
poly_cos_prec_list = [self.precision] * len(poly_degree_cos_list)
# sine polynomial: limiting first coefficient precision to 1-bit
poly_sin_prec_list = [self.precision] * len(poly_degree_sin_list)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
Log.report(Log.Info,
"building mathematical polynomials for sin and cos")
# Polynomial approximations
Log.report(Log.Info, "cos")
poly_object_cos, poly_error_cos = Polynomial.build_from_approximation_with_error(
cos(sollya.x),
poly_degree_cos_list,
poly_cos_prec_list,
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(Log.Info, "sin")
poly_object_sin, poly_error_sin = Polynomial.build_from_approximation_with_error(
sin(sollya.x),
poly_degree_sin_list,
poly_sin_prec_list,
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(
Log.Info, "poly error cos: {} / {:d}".format(
poly_error_cos, int(sollya.log2(poly_error_cos))))
Log.report(
Log.Info, "poly error sin: {0} / {1:d}".format(
poly_error_sin, int(sollya.log2(poly_error_sin))))
Log.report(Log.Info, "poly cos : %s" % poly_object_cos)
Log.report(Log.Info, "poly sin : %s" % poly_object_sin)
# Polynomial evaluation scheme
poly_cos = polynomial_scheme_builder(
poly_object_cos.sub_poly(start_index=1),
red_vx,
unified_precision=self.precision)
poly_sin = polynomial_scheme_builder(
poly_object_sin.sub_poly(start_index=2),
red_vx,
unified_precision=self.precision)
poly_cos.set_attributes(tag="poly_cos", debug=debug_multi)
poly_sin.set_attributes(tag="poly_sin",
debug=debug_multi,
unbreakable=True)
# TwoProductFMA
mul_cos_x = tabulated_cos * poly_cos
mul_cos_y = FusedMultiplyAdd(tabulated_cos,
poly_cos,
-mul_cos_x,
precision=self.precision)
mul_sin_x = tabulated_sin * poly_sin
mul_sin_y = FusedMultiplyAdd(tabulated_sin,
poly_sin,
-mul_sin_x,
precision=self.precision)
mul_coeff_sin_hi = tabulated_sin * red_vx
mul_coeff_sin_lo = FusedMultiplyAdd(tabulated_sin, red_vx,
-mul_coeff_sin_hi)
mul_cos = Addition(mul_cos_x,
mul_cos_y,
precision=self.precision,
tag="mul_cos") #, debug = debug_multi)
mul_sin = Negation(Addition(mul_sin_x,
mul_sin_y,
precision=self.precision),
precision=self.precision,
tag="mul_sin") #, debug = debug_multi)
mul_coeff_sin = Negation(Addition(mul_coeff_sin_hi,
mul_coeff_sin_lo,
precision=self.precision),
precision=self.precision,
tag="mul_coeff_sin") #, debug = debug_multi)
mul_cos_x.set_attributes(
tag="mul_cos_x", precision=self.precision) #, debug = debug_multi)
mul_cos_y.set_attributes(
tag="mul_cos_y", precision=self.precision) #, debug = debug_multi)
mul_sin_x.set_attributes(
tag="mul_sin_x", precision=self.precision) #, debug = debug_multi)
mul_sin_y.set_attributes(
tag="mul_sin_y", precision=self.precision) #, debug = debug_multi)
cos_eval_d_1 = (((mul_cos + mul_sin) + mul_coeff_sin) + tabulated_cos)
cos_eval_d_1.set_attributes(tag="cos_eval_d_1",
precision=self.precision,
debug=debug_multi)
result_1 = Statement(Return(cos_eval_d_1))
#######################################################################
# LARGE ARGUMENT MANAGEMENT #
# (lar: Large Argument Reduction) #
#######################################################################
# payne and hanek argument reduction for large arguments
ph_k = frac_pi_index
ph_frac_pi = round(S2**ph_k / pi, 1500, sollya.RN)
ph_inv_frac_pi = pi / S2**ph_k
ph_statement, ph_acc, ph_acc_int = generate_payne_hanek(vx,
ph_frac_pi,
self.precision,
n=100,
k=ph_k)
# assigning Large Argument Reduction reduced variable
lar_vx = Variable("lar_vx",
precision=self.precision,
var_type=Variable.Local)
lar_red_vx = Addition(Multiplication(lar_vx,
inv_frac_pi,
precision=self.precision),
Multiplication(lar_vx,
inv_frac_pi_lo,
precision=self.precision),
precision=self.precision,
tag="lar_red_vx",
debug=debug_multi)
C32 = Constant(2**(ph_k + 1), precision=int_precision, tag="C32")
ph_acc_int_red = Select(ph_acc_int < C0,
C32 + ph_acc_int,
ph_acc_int,
precision=int_precision,
tag="ph_acc_int_red")
if self.sin_output:
lar_offset_k = Addition(ph_acc_int_red,
C_offset,
precision=int_precision,
tag="lar_offset_k")
else:
lar_offset_k = ph_acc_int_red
ph_acc_int_red.set_attributes(tag="ph_acc_int_red", debug=debug_multi)
lar_modk = BitLogicAnd(lar_offset_k,
2**(frac_pi_index + 1) - 1,
precision=int_precision,
tag="lar_modk",
debug=debug_multi)
lar_statement = Statement(ph_statement,
ReferenceAssign(lar_vx,
ph_acc,
debug=debug_multi),
ReferenceAssign(red_vx,
lar_red_vx,
debug=debug_multi),
ReferenceAssign(modk, lar_modk),
prevent_optimization=True)
test_NaN_or_Inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
tag="NaN_or_Inf",
debug=debug_multi)
return_NaN_or_Inf = Statement(Return(FP_QNaN(self.precision)))
scheme = ConditionBlock(
test_NaN_or_Inf, Statement(ClearException(), return_NaN_or_Inf),
Statement(
modk, red_vx,
ConditionBlock(
test_ph_bound, lar_statement,
Statement(
ReferenceAssign(modk, modk_std),
ReferenceAssign(red_vx, red_vx_std),
)), result_1))
return scheme
def generic_atan2_generate(self, _vx, vy=None):
""" if vy is None, compute atan(_vx), else compute atan2(vy / vx) """
if vy is None:
# approximation
# if abs_vx <= 1.0 then atan(abx_vx) is directly approximated
# if abs_vx > 1.0 then atan(abs_vx) = pi/2 - atan(1 / abs_vx)
#
# for vx >= 0, atan(vx) = atan(abs_vx)
#
# for vx < 0, atan(vx) = -atan(abs_vx) for vx < 0
# = -pi/2 + atan(1 / abs_vx)
vx = _vx
sign_cond = vx < 0
abs_vx = Select(vx < 0, -vx, vx, tag="abs_vx", debug=debug_multi)
bound_cond = abs_vx > 1
inv_abs_vx = 1 / abs_vx
# condition to select subtraction
cond = LogicalOr(LogicalAnd(vx < 0, LogicalNot(bound_cond)),
vx > 1,
tag="cond",
debug=debug_multi)
# reduced argument
red_vx = Select(bound_cond,
inv_abs_vx,
abs_vx,
tag="red_vx",
debug=debug_multi)
offset = None
else:
# bound_cond is True iff Abs(vy / _vx) > 1.0
bound_cond = Abs(vy) > Abs(_vx)
bound_cond.set_attributes(tag="bound_cond", debug=debug_multi)
# vx and vy are of opposite signs
#sign_cond = (_vx * vy) < 0
# using cast to int(signed) and bitwise xor
# to determine if _vx and vy are of opposite sign rapidly
fast_sign_cond = BitLogicXor(
TypeCast(_vx, precision=self.precision.get_integer_format()),
TypeCast(vy, precision=self.precision.get_integer_format()),
precision=self.precision.get_integer_format()) < 0
# sign_cond = (_vx * vy) < 0
sign_cond = fast_sign_cond
sign_cond.set_attributes(tag="sign_cond", debug=debug_multi)
# condition to select subtraction
# TODO: could be accelerated if LogicalXor existed
slow_cond = LogicalOr(
LogicalAnd(sign_cond,
LogicalNot(bound_cond)), # 1 < (vy / _vx) < 0
LogicalAnd(bound_cond,
LogicalNot(sign_cond)), # (vy / _vx) > 1
tag="cond",
debug=debug_multi)
cond = slow_cond
numerator = Select(bound_cond,
_vx,
vy,
tag="numerator",
debug=debug_multi)
denominator = Select(bound_cond,
vy,
_vx,
tag="denominator",
debug=debug_multi)
# reduced argument
red_vx = Abs(numerator) / Abs(denominator)
red_vx.set_attributes(tag="red_vx", debug=debug_multi)
offset = Select(
_vx > 0,
Constant(0, precision=self.precision),
# vx < 0
Select(
sign_cond,
# vy > 0
Constant(sollya.pi, precision=self.precision),
Constant(-sollya.pi, precision=self.precision),
precision=self.precision),
precision=self.precision,
tag="offset")
approx_fct = sollya.atan(sollya.x)
if self.method == "piecewise":
sign_vx = Select(cond,
-1,
1,
precision=self.precision,
tag="sign_vx",
debug=debug_multi)
cst_sign = Select(sign_cond,
-1,
1,
precision=self.precision,
tag="cst_sign",
debug=debug_multi)
cst = cst_sign * Select(
bound_cond, sollya.pi / 2, 0, precision=self.precision)
cst.set_attributes(tag="cst", debug=debug_multi)
bound_low = 0.0
bound_high = 1.0
num_intervals = self.num_sub_intervals
error_threshold = S2**-(self.precision.get_mantissa_size() + 8)
approx, eval_error = piecewise_approximation(
approx_fct,
red_vx,
self.precision,
bound_low=bound_low,
bound_high=bound_high,
max_degree=None,
num_intervals=num_intervals,
error_threshold=error_threshold,
odd=True)
result = cst + sign_vx * approx
result.set_attributes(tag="result",
precision=self.precision,
debug=debug_multi)
elif self.method == "single":
approx_interval = Interval(0, 1.0)
# determining the degree of the polynomial approximation
poly_degree_range = sollya.guessdegree(
approx_fct / sollya.x, approx_interval,
S2**-(self.precision.get_field_size() + 2))
poly_degree = int(sollya.sup(poly_degree_range)) + 4
Log.report(Log.Info, "poly_degree={}".format(poly_degree))
# arctan is an odd function, so only odd coefficient must be non-zero
poly_degree_list = list(range(1, poly_degree + 1, 2))
poly_object, poly_error = Polynomial.build_from_approximation_with_error(
approx_fct, poly_degree_list,
[1] + [self.precision.get_sollya_object()] *
(len(poly_degree_list) - 1), approx_interval)
odd_predicate = lambda index, _: ((index - 1) % 4 != 0)
even_predicate = lambda index, _: (index != 1 and
(index - 1) % 4 == 0)
poly_odd_object = poly_object.sub_poly_cond(odd_predicate,
offset=1)
poly_even_object = poly_object.sub_poly_cond(even_predicate,
offset=1)
sollya.settings.display = sollya.hexadecimal
Log.report(Log.Info, "poly_error: {}".format(poly_error))
Log.report(Log.Info, "poly_odd: {}".format(poly_odd_object))
Log.report(Log.Info, "poly_even: {}".format(poly_even_object))
poly_odd = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_odd_object, abs_vx)
poly_odd.set_attributes(tag="poly_odd", debug=debug_multi)
poly_even = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_even_object, abs_vx)
poly_even.set_attributes(tag="poly_even", debug=debug_multi)
exact_sum = poly_odd + poly_even
exact_sum.set_attributes(tag="exact_sum", debug=debug_multi)
# poly_even should be (1 + poly_even)
result = vx + vx * exact_sum
result.set_attributes(tag="result",
precision=self.precision,
debug=debug_multi)
else:
raise NotImplementedError
if not offset is None:
result = result + offset
std_scheme = Statement(Return(result))
scheme = std_scheme
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.get_input_precision().sollya_object
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# 2-limb approximation of log(2)
# hi part precision is reduced to provide exact operation
# when multiplied by an exponent value
log2_hi_value = round(log(2), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
int_precision = self.precision.get_integer_format()
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision = self.precision)
dummy_rcp_seed = ReciprocalSeed(dummy_var, precision = self.precision)
inv_approx_table = self.processor.get_recursive_implementation(dummy_rcp_seed, language = None, table_getter = lambda self: self.approx_table_map)
# table creation
table_index_size = inv_approx_table.index_size
log_table = ML_NewTable(dimensions = [2**table_index_size, 2], storage_precision = self.precision)
# storing accurate logarithm approximation of value returned
# by the fast reciprocal operation
for i in range(0, 2**table_index_size):
inv_value = inv_approx_table[i]
value_high = round(log(inv_value), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
neg_input = Comparison(vx, -1, likely=False, precision=ML_Bool, specifier=Comparison.Less, debug=debug_multi, tag="neg_input")
vx_nan_or_inf = Test(vx, specifier=Test.IsInfOrNaN, likely=False, precision=ML_Bool, debug=debug_multi, tag="nan_or_inf")
vx_snan = Test(vx, specifier=Test.IsSignalingNaN, likely=False, debug=debug_multi, tag="snan")
vx_inf = Test(vx, specifier=Test.IsInfty, likely=False, debug=debug_multi, tag="inf")
vx_subnormal = Test(vx, specifier=Test.IsSubnormal, likely=False, debug=debug_multi, tag="vx_subnormal")
# for x = m.2^e, such that e >= 0
#
# log(1+x) = log(1 + m.2^e)
# = log(2^e . 2^-e + m.2^e)
# = log(2^e . (2^-e + m))
# = log(2^e) + log(2^-e + m)
# = e . log(2) + log (2^-e + m)
#
# t = (2^-e + m)
# t = m_t . 2^e_t
# r ~ 1 / m_t => r.m_t ~ 1 ~ 0
#
# t' = t . 2^-e_t
# = 2^-e-e_t + m . 2^-e_t
#
# if e >= 0, then 2^-e <= 1, then 1 <= m + 2^-e <= 3
# r = m_r . 2^e_r
#
# log(1+x) = e.log(2) + log(r . 2^e_t . 2^-e_t . (2^-e + m) / r)
# = e.log(2) + log(r . 2^(-e-e_t) + r.m.2^-e_t) + e_t . log(2)- log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + P_log1p(r . t' - 1) - log(r)
#
#
# argument reduction
m = MantissaExtraction(vx, tag="vx", precision=self.precision, debug=debug_multi)
e = ExponentExtraction(vx, tag="e", precision=int_precision, debug=debug_multi)
# 2^-e
TwoMinusE = ExponentInsertion(-e, tag="Two_minus_e", precision=self.precision, debug=debug_multi)
t = Addition(TwoMinusE, m, precision=self.precision, tag="t", debug=debug_multi)
m_t = MantissaExtraction(t, tag="m_t", precision=self.precision, debug=debug_multi)
e_t = ExponentExtraction(t, tag="e_t", precision=int_precision, debug=debug_multi)
# 2^(-e-e_t)
TwoMinusEEt = ExponentInsertion(-e-e_t, tag="Two_minus_e_et", precision=self.precision)
TwoMinusEt = ExponentInsertion(-e_t, tag="Two_minus_et", precision=self.precision, debug=debug_multi)
rcp_mt = ReciprocalSeed(m_t, tag="rcp_mt", precision=self.precision, debug=debug_multi)
INDEX_SIZE = table_index_size
table_index = generic_mantissa_msb_index_fct(INDEX_SIZE, m_t)
table_index.set_attributes(tag="table_index", debug=debug_multi)
log_inv_lo = TableLoad(log_table, table_index, 1, tag="log_inv_lo", debug=debug_multi)
log_inv_hi = TableLoad(log_table, table_index, 0, tag="log_inv_hi", debug=debug_multi)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
approx_fct = sollya.log1p(sollya.x) / (sollya.x)
poly_degree = sup(guessdegree(approx_fct, approx_interval, S2**-(self.precision.get_field_size()+1))) + 1
Log.report(Log.Debug, "poly_degree is {}", poly_degree)
global_poly_object = Polynomial.build_from_approximation(approx_fct, poly_degree, [self.precision]*(poly_degree+1), approx_interval, sollya.absolute)
poly_object = global_poly_object # .sub_poly(start_index=1)
EXT_PRECISION_MAP = {
ML_Binary32: ML_SingleSingle,
ML_Binary64: ML_DoubleDouble,
ML_SingleSingle: ML_TripleSingle,
ML_DoubleDouble: ML_TripleDouble
}
if not self.precision in EXT_PRECISION_MAP:
Log.report(Log.Error, "no extended precision available for {}", self.precision)
ext_precision = EXT_PRECISION_MAP[self.precision]
# pre_rtp = r . 2^(-e-e_t) + m .2^-e_t
pre_rtp = Addition(
rcp_mt * TwoMinusEEt,
Multiplication(
rcp_mt,
Multiplication(
m,
TwoMinusEt,
precision=self.precision,
tag="pre_mult",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_mult2",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_rtp",
debug=debug_multi
)
pre_red_vx = Addition(
pre_rtp,
-1,
precision=ext_precision,
)
red_vx = Conversion(pre_red_vx, precision=self.precision, tag="red_vx", debug=debug_multi)
Log.report(Log.Info, "generating polynomial evaluation scheme")
poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vx, unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Debug, "{}", global_poly_object.get_sollya_object())
fp_e = Conversion(e + e_t, precision=self.precision, tag="fp_e", debug=debug_multi)
ext_poly = Multiplication(red_vx, poly, precision=ext_precision)
pre_result = Addition(
Addition(
fp_e * log2_hi,
fp_e * log2_lo,
precision=ext_precision
),
Addition(
Addition(
-log_inv_hi,
-log_inv_lo,
precision=ext_precision
),
ext_poly,
precision=ext_precision
),
precision=ext_precision
)
result = Conversion(pre_result, precision=self.precision, tag="result", debug=debug_multi)
# main scheme
Log.report(Log.Info, "MDL scheme")
pre_scheme = ConditionBlock(neg_input,
Statement(
ClearException(),
Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))
),
ConditionBlock(vx_nan_or_inf,
ConditionBlock(vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(
ClearException(),
ConditionBlock(vx_snan,
Raise(ML_FPE_Invalid)
),
Return(FP_QNaN(self.precision))
)
),
Return(result)
)
)
scheme = pre_scheme
return scheme
def generate_scheme(self):
"""Produce an abstract scheme for the logarithm.
This abstract scheme will be used by the code generation backend.
"""
if self.precision not in [ML_Binary32, ML_Binary64]:
Log.report(Log.Error, "The demanded precision is not supported")
vx = self.implementation.add_input_variable("x", self.precision)
def default_bool_convert(optree, precision=None, **kw):
return bool_convert(optree, precision, -1, 0, **kw) \
if isinstance(self.processor, VectorBackend) \
else bool_convert(optree, precision, 1, 0, **kw)
precision = self.precision.sollya_object
int_prec = self.precision.get_integer_format()
Log.report(Log.Info, "int_prec is %s" % int_prec)
uint_prec = self.precision.get_unsigned_integer_format()
Log.report(Log.Info, "MDL constants")
cgpe_scheme_idx = int(self.cgpe_index)
table_index_size = int(self.tbl_index_size)
#
table_nb_elements = 2**(table_index_size)
table_dimensions = [2*table_nb_elements] # two values are stored for each element
field_size = Constant(self.precision.get_field_size(),
precision = int_prec,
tag = 'field_size')
if self.log_radix == EXP_1:
log2_hi = Constant(
round(log(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log(2) - round(log(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
elif self.log_radix == 10:
log2_hi = Constant(
round(log10(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log10(2) - round(log10(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
# ... if log_radix == '2' then log2(2) == 1
# subnormal_mask aims at trapping positive subnormals except zero.
# That's why we will subtract 1 to the integer bitstring of the input, and
# then compare for Less (strict) the resulting integer bitstring to this
# mask, e.g. 0x7fffff for binary32.
if self.no_subnormal == False:
subnormal_mask = Constant((1 << self.precision.get_field_size()) - 1,
precision = int_prec, tag = 'subnormal_mask')
fp_one = Constant(1.0, precision = self.precision, tag = 'fp_one')
fp_one_as_uint = TypeCast(fp_one, precision = uint_prec,
tag = 'fp_one_as_uint')
int_zero = Constant(0, precision = int_prec, tag = 'int_zero')
int_one = Constant(1, precision = int_prec, tag = 'int_one')
table_mantissa_half_ulp = Constant(
1 << (self.precision.field_size - table_index_size - 1),
precision = int_prec
)
table_s_exp_index_mask = Constant(
~((table_mantissa_half_ulp.get_value() << 1) - 1),
precision = uint_prec
)
Log.report(Log.Info, "MDL table")
# The table holds approximations of -log(2^tau * r_i) so we first compute
# the index value for which tau changes from 1 to 0.
cut = sqrt(2.)
tau_index_limit = floor(table_nb_elements * (2./cut - 1))
sollya_logtbl = [
(-log1p(float(i) / table_nb_elements)
+ (0 if i <= tau_index_limit else log(2.))) / log(self.log_radix)
for i in range(table_nb_elements)
]
# ...
init_logtbl_hi = [
round(sollya_logtbl[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl_lo = [
round(sollya_logtbl[i] - init_logtbl_hi[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl = [tmp[i] for i in range(len(init_logtbl_hi)) for tmp in [init_logtbl_hi, init_logtbl_lo]]
log1p_table = ML_NewTable(dimensions = table_dimensions,
storage_precision = self.precision,
init_data = init_logtbl,
tag = 'ml_log1p_table')
# ...
if self.no_rcp:
sollya_rcptbl = [
(1/((1+float(i)/table_nb_elements)+2**(-1-int(self.tbl_index_size))))
for i in range(table_nb_elements)
]
init_rcptbl = [
round(sollya_rcptbl[i],
int(self.tbl_index_size)+1, # self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
rcp_table = ML_NewTable(dimensions = [table_nb_elements],
storage_precision = self.precision,
init_data = init_rcptbl,
tag = 'ml_rcp_table')
# ...
Log.report(Log.Info, 'MDL unified subnormal handling')
vx_as_int = TypeCast(vx, precision = int_prec, tag = 'vx_as_int')
if self.no_subnormal == False:
vx_as_uint = TypeCast(vx, precision = uint_prec, tag = 'vx_as_uint')
# Avoid the 0.0 case by subtracting 1 from vx_as_int
tmp = Comparison(vx_as_int - 1, subnormal_mask,
specifier = Comparison.Less)
is_subnormal = default_bool_convert(
tmp, # Will catch negative values as well as NaNs with sign bit set
precision = int_prec)
is_subnormal.set_attributes(tag = "is_subnormal")
if not(isinstance(self.processor, VectorBackend)):
is_subnormal = Subtraction(Constant(0, precision = int_prec),
is_subnormal,
precision = int_prec)
#################################################
# Vectorizable integer based subnormal handling #
#################################################
# 1. lzcnt
# custom lzcount-like for subnormal numbers using FPU (see draft article)
Zi = BitLogicOr(vx_as_uint, fp_one_as_uint, precision = uint_prec, tag="Zi")
Zf = Subtraction(
TypeCast(Zi, precision = self.precision),
fp_one,
precision = self.precision,
tag="Zf")
# Zf exponent is -(nlz(x) - exponent_size).
# 2. compute shift value
# Vectorial comparison on x86+sse/avx is going to look like
# '|0x00|0xff|0x00|0x00|' and that's why we use Negate.
# But for scalar code generation, comparison will rather be either 0 or 1
# in C. Thus mask below won't be correct for a scalar implementation.
# FIXME: Can we know the backend that will be called and choose in
# consequence? Should we make something arch-agnostic instead?
#
n_value = BitLogicAnd(
Addition(
DirtyExponentExtraction(Zf, self.precision),
Constant(
self.precision.get_bias(),
precision = int_prec),
precision = int_prec),
is_subnormal,
precision = int_prec,
tag = "n_value")
alpha = Negation(n_value, tag="alpha")
#
# 3. shift left
# renormalized_mantissa = BitLogicLeftShift(vx_as_int, value)
normal_vx_as_int = BitLogicLeftShift(vx_as_int, alpha)
# 4. set exponent to the right value
# Compute the exponent to add : (p-1)-(value) + 1 = p-1-value
# The final "+ 1" comes from the fact that once renormalized, the
# floating-point datum has a biased exponent of 1
#tmp0 = Subtraction(
# field_size,
# value,
# precision = int_prec,
# tag="tmp0")
# Set the value to 0 if the number is not subnormal
#tmp1 = BitLogicAnd(tmp0, is_subnormal)
#renormalized_exponent = BitLogicLeftShift(
# tmp1,
# field_size
# )
else: # no_subnormal == True
normal_vx_as_int = vx_as_int
#normal_vx_as_int = renormalized_mantissa + renormalized_exponent
normal_vx = TypeCast(normal_vx_as_int, precision = self.precision,
tag = 'normal_vx')
# alpha = BitLogicAnd(field_size, is_subnormal, tag = 'alpha')
# XXX Extract the mantissa, see if this is supported in the x86 vector
# backend or if it still uses the support_lib.
vx_mantissa = MantissaExtraction(normal_vx, precision = self.precision)
Log.report(Log.Info, "MDL scheme")
if self.force_division == True:
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
elif self.no_rcp == False:
rcp_m = ReciprocalSeed(vx_mantissa, precision = self.precision)
if not self.processor.is_supported_operation(rcp_m):
if self.precision == ML_Binary64:
# Try using a binary32 FastReciprocal
binary32_m = Conversion(vx_mantissa, precision = ML_Binary32)
rcp_m = ReciprocalSeed(binary32_m, precision = ML_Binary32)
rcp_m = Conversion(rcp_m, precision = ML_Binary64)
if not self.processor.is_supported_operation(rcp_m):
# FIXME An approximation table could be used instead but for vector
# implementations another GATHER would be required.
# However this may well be better than a division...
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
else: # ... use a look-up table
rcp_shift = BitLogicLeftShift(normal_vx_as_int, self.precision.get_exponent_size() + 1)
rcp_idx = BitLogicRightShift(rcp_shift, self.precision.get_exponent_size() + 1 + self.precision.get_field_size() - int(self.tbl_index_size))
rcp_m = TableLoad(rcp_table, rcp_idx, tag = 'rcp_idx',
debug = debug_multi)
#
rcp_m.set_attributes(tag = 'rcp_m')
# exponent is normally either 0 or -1, since m is in [1, 2). Possible
# optimization?
# exponent = ExponentExtraction(rcp_m, precision = self.precision,
# tag = 'exponent')
ri_round = TypeCast(
Addition(
TypeCast(rcp_m, precision = int_prec),
table_mantissa_half_ulp,
precision = int_prec
),
precision = uint_prec
)
ri_fast_rndn = BitLogicAnd(
ri_round,
table_s_exp_index_mask,
tag = 'ri_fast_rndn',
precision = uint_prec
)
# u = m * ri - 1
ul = None
if self.no_rcp == True: # ... u does not fit on a single word
tmp_u, tmp_ul = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = (self.no_fma == False))
fp_minus_one = Constant(-1.0, precision = self.precision, tag = 'fp_minus_one')
u, ul = Add212(fp_minus_one, tmp_u, tmp_ul)
u.set_attributes(tag='uh')
ul.set_attributes(tag='ul')
elif self.no_fma == False:
u = FusedMultiplyAdd(
vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fp_one,
specifier = FusedMultiplyAdd.Subtract,
tag = 'u')
else: # disable FMA
# tmph + tmpl = m * ri, where tmph ~ 1
tmph, tmpl = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = False)
# u_tmp = tmph - 1 ... exact due to Sterbenz
u_tmp = Subtraction(tmph, fp_one, precision = self.precision)
# u = u_tmp - tmpl ... exact since the result u is representable as a single word
u = Addition(u_tmp, tmpl, precision = self.precision, tag = 'u')
unneeded_bits = Constant(
self.precision.field_size - table_index_size,
precision=uint_prec,
tag="unneeded_bits"
)
assert self.precision.field_size - table_index_size >= 0
ri_bits = BitLogicRightShift(
ri_fast_rndn,
unneeded_bits,
precision = uint_prec,
tag = "ri_bits"
)
# Retrieve mantissa's MSBs + first bit of exponent, for tau computation in case
# exponent is 0 (i.e. biased 127, i.e. first bit of exponent is set.).
# In this particular case, i = 0 but tau is 1
# table_index does not need to be as long as uint_prec might be,
# try and keep it the size of size_t.
size_t_prec = ML_UInt32
signed_size_t_prec = ML_Int32
table_index_mask = Constant(
(1 << (table_index_size + 1)) - 1,
precision = size_t_prec
)
table_index = BitLogicAnd(
Conversion(ri_bits, precision = size_t_prec),
table_index_mask,
tag = 'table_index',
precision = size_t_prec
)
# Compute tau using the tau_index_limit value.
tmp = default_bool_convert(
Comparison(
TypeCast(table_index, precision = signed_size_t_prec),
Constant(tau_index_limit, precision = signed_size_t_prec),
specifier = Comparison.Greater
if isinstance(self.processor, VectorBackend)
else Comparison.LessOrEqual
),
precision = signed_size_t_prec,
tag="tmp"
)
# A true tmp will typically be -1 for VectorBackends, but 1 for standard C.
tau = Conversion(
Addition(tmp, Constant(1, precision=signed_size_t_prec), precision = signed_size_t_prec, tag="pre_add")
if isinstance(self.processor, VectorBackend)
else tmp,
precision=int_prec,
tag="pre_tau"
)
tau.set_attributes(tag = 'tau')
# Update table_index: keep only table_index_size bits
table_index_hi = BitLogicAnd(
table_index,
Constant((1 << table_index_size) - 1, precision = size_t_prec),
precision = size_t_prec
)
# table_index_hi = table_index_hi << 1
table_index_hi = BitLogicLeftShift(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_hi"
)
# table_index_lo = table_index_hi + 1
table_index_lo = Addition(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_lo"
)
tbl_hi = TableLoad(log1p_table, table_index_hi, tag = 'tbl_hi',
debug = debug_multi)
tbl_lo = TableLoad(log1p_table, table_index_lo, tag = 'tbl_lo',
debug = debug_multi)
# Compute exponent e + tau - alpha, but first subtract the bias.
if self.no_subnormal == False:
tmp_eptau = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
exponent = Subtraction(tmp_eptau, alpha, precision = int_prec)
else:
exponent = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
#
fp_exponent = Conversion(exponent, precision = self.precision,
tag = 'fp_exponent')
Log.report(Log.Info, 'MDL polynomial approximation')
if self.log_radix == EXP_1:
sollya_function = log(1 + sollya.x)
elif self.log_radix == 2:
sollya_function = log2(1 + sollya.x)
elif self.log_radix == 10:
sollya_function = log10(1 + sollya.x)
# ...
if self.force_division == True: # rcp accuracy is 2^(-p)
boundrcp = 2**(-self.precision.get_precision())
else:
boundrcp = 1.5 * 2**(-12) # ... see Intel intrinsics guide
if self.precision in [ML_Binary64]:
if not self.processor.is_supported_operation(rcp_m):
boundrcp = (1+boundrcp)*(1+2**(-24)) - 1
else:
boundrcp = 2**(-14) # ... see Intel intrinsics guide
arg_red_mag = boundrcp + 2**(-table_index_size-1) + boundrcp * 2**(-table_index_size-1)
if self.no_rcp == False:
approx_interval = Interval(-arg_red_mag, arg_red_mag)
else:
approx_interval = Interval(-2**(-int(self.tbl_index_size)+1),2**(-int(self.tbl_index_size)+1))
max_eps = 2**-(2*(self.precision.get_field_size()))
Log.report(Log.Info, "max acceptable error for polynomial = {}".format(float.hex(max_eps)))
poly_degree = sup(
guessdegree(
sollya_function,
approx_interval,
max_eps,
)
)
Log.report(Log.Info, "poly degree is ", poly_degree)
if self.log_radix == EXP_1:
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(2, int(poly_degree) + 1), # Force 1st 2 coeffs to 0 and 1, resp.
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree - 1),
approx_interval,
sollya.absolute,
0 + sollya._x_) # Force the first 2 coefficients to 0 and 1, resp.
else: # ... == '2' or '10'
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(1, int(poly_degree) + 1), # Force 1st coeff to 0
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree),
approx_interval,
sollya.absolute,
0) # Force the first coefficients to 0
Log.report(Log.Info, str(poly_object))
constant_precision = ML_SingleSingle if self.precision == ML_Binary32 \
else ML_DoubleDouble if self.precision == ML_Binary64 \
else None
if is_cgpe_available():
log1pu_poly = PolynomialSchemeEvaluator.generate_cgpe_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision, scheme_id = cgpe_scheme_idx
)
else:
Log.report(Log.Warning,
"CGPE not available, falling back to std poly evaluator")
log1pu_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision
)
# XXX Dirty implementation of double-(self.precision) poly
def dirty_poly_node_conversion(node, variable_h, variable_l, use_fma):
return dirty_multi_node_expand(
node, self.precision, mem_map={variable_h: (variable_h, variable_l)}, fma=use_fma)
log1pu_poly_hi, log1pu_poly_lo = dirty_poly_node_conversion(log1pu_poly, u, ul,
use_fma=(self.no_fma == False))
log1pu_poly_hi.set_attributes(tag = 'log1pu_poly_hi')
log1pu_poly_lo.set_attributes(tag = 'log1pu_poly_lo')
# Compute log(2) * (e + tau - alpha)
if self.log_radix != 2: # 'e' or '10'
log2e_hi, log2e_lo = Mul212(fp_exponent, log2_hi, log2_lo,
fma = (self.no_fma == False))
# Add log1p(u)
if self.log_radix != 2: # 'e' or '10'
tmp_res_hi, tmp_res_lo = Add222(log2e_hi, log2e_lo,
log1pu_poly_hi, log1pu_poly_lo)
else:
tmp_res_hi, tmp_res_lo = Add212(fp_exponent,
log1pu_poly_hi, log1pu_poly_lo)
# Add -log(2^(tau)/m) approximation retrieved by two table lookups
logx_hi = Add122(tmp_res_hi, tmp_res_lo, tbl_hi, tbl_lo)[0]
logx_hi.set_attributes(tag = 'logx_hi')
scheme = Return(logx_hi, precision = self.precision)
return scheme
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_multi)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debug_multi)
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debug_multi)
tho_cond = _vx_mant > Constant(sollya.sqrt(2),
precision=self.precision)
tho = Select(tho_cond,
Constant(1.0, precision=self.precision),
Constant(0.0, precision=self.precision),
precision=self.precision,
tag="tho",
debug=debug_multi)
rcp = ReciprocalSeed(_vx_mant, precision=self.precision, tag="rcp")
r = Multiplication(rcp,
_vx_mant,
precision=self.precision,
tag="r")
int_format = self.precision.get_integer_format()
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=int_format),
Constant(-2, precision=int_format),
precision=int_format),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_multi)
_red_vx = arg_red_index * _vx_mant - 1.0
inv_err = S2**-6
red_interval = Interval(1 - inv_err, 1 + inv_err)
_red_vx.set_attributes(tag="_red_vx",
debug=debug_multi,
interval=red_interval)
# return in case of standard (non-special) input
_log_inv_lo = Select(tho_cond,
TableLoad(log_table_tho, table_index, 1),
TableLoad(log_table, table_index, 1),
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = Select(tho_cond,
TableLoad(log_table_tho, table_index, 0),
TableLoad(log_table, table_index, 0),
tag="log_inv_hi",
debug=debug_multi)
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + x) / x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Info, "generating polynomial evaluation scheme")
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Info, poly_object.get_sollya_object())
corr_exp = Conversion(_vx_exp if exp_corr_factor == None else
_vx_exp + exp_corr_factor,
precision=self.precision) + tho
corr_exp.set_attributes(tag="corr_exp", debug=debug_multi)
# _poly approximates log10(1+r)/r
# _poly * red_vx approximates log10(x)
m0h, m0l = Mul211(_red_vx, _poly)
m0h, m0l = Add212(_red_vx, m0h, m0l)
m0h.set_attributes(tag="m0h", debug=debug_multi)
m0l.set_attributes(tag="m0l")
l0_h = corr_exp * log2_hi
l0_l = corr_exp * log2_lo
l0_h.set_attributes(tag="l0_h")
l0_l.set_attributes(tag="l0_l")
rh, rl = Add222(l0_h, l0_l, m0h, m0l)
rh.set_attributes(tag="rh0", debug=debug_multi)
rl.set_attributes(tag="rl0", debug=debug_multi)
rh, rl = Add222(-_log_inv_hi, -_log_inv_lo, rh, rl)
rh.set_attributes(tag="rh", debug=debug_multi)
rl.set_attributes(tag="rl", debug=debug_multi)
if sollya.log(self.basis) != 1.0:
lbh = self.precision.round_sollya_object(
1 / sollya.log(self.basis))
lbl = self.precision.round_sollya_object(
1 / sollya.log(self.basis) - lbh)
rh, rl = Mul222(rh, rl, lbh, lbl)
return rh
else:
return rh
def generate_scheme(self):
""" generate scheme """
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
# retrieving processor inverse approximation table
lo_bound_global = SollyaObject(0.0)
hi_bound_global = SollyaObject(0.75)
approx_interval = Interval(lo_bound_global, hi_bound_global)
approx_interval_size = hi_bound_global - lo_bound_global
# table creation
table_index_size = 7
field_index_size = 2
exp_index_size = table_index_size - field_index_size
table_size = 2**table_index_size
table_index_range = range(table_size)
local_degree = 9
coeff_table = ML_NewTable(dimensions=[table_size, local_degree],
storage_precision=self.precision)
exp_lo = 2**exp_index_size
for i in table_index_range:
lo_bound = (1.0 + (i % 2**field_index_size) * S2**-field_index_size
) * S2**(i / 2**field_index_size - exp_lo)
hi_bound = (1.0 +
((i % 2**field_index_size) + 1) * S2**-field_index_size
) * S2**(i / 2**field_index_size - exp_lo)
local_approx_interval = Interval(lo_bound, hi_bound)
local_poly_object, local_error = Polynomial.build_from_approximation_with_error(
acos(1 - sollya.x), local_degree,
[self.precision] * (local_degree + 1), local_approx_interval,
sollya.absolute)
local_error = int(
log2(sup(abs(local_error / acos(1 - local_approx_interval)))))
coeff_table
for d in range(local_degree):
coeff_table[i][d] = sollya.coeff(
local_poly_object.get_sollya_object(), d)
table_index = BitLogicRightShift(
vx,
vx.get_precision().get_field_size() -
field_index_size) - (exp_lo << field_index_size)
print "building mathematical polynomial"
poly_degree = sup(
sollya.guessdegree(acos(x), approx_interval,
S2**-(self.precision.get_field_size())))
print "guessed polynomial degree: ", int(poly_degree)
#global_poly_object = Polynomial.build_from_approximation(log10(1+x)/x, poly_degree, [self.precision]*(poly_degree+1), approx_interval, absolute)
print "generating polynomial evaluation scheme"
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
# building eval error map
#eval_error_map = {
# red_vx: Variable("red_vx", precision = self.precision, interval = red_vx.get_interval()),
# log_inv_hi: Variable("log_inv_hi", precision = self.precision, interval = table_high_interval),
# log_inv_lo: Variable("log_inv_lo", precision = self.precision, interval = table_low_interval),
#}
# computing gappa error
#poly_eval_error = self.get_eval_error(result, eval_error_map)
# main scheme
print "MDL scheme"
scheme = Statement(Return(vx))
return scheme
def generate_reduced_log_split(self,
_vx_mant,
log_f,
inv_approx_table,
log_table,
log_table_tho=None,
corr_exp=None,
tho_cond=None):
""" Generate a logarithm approximation (log_f(_vx_mant) + corr_exp) for a reduced argument
_vx_mant which is assumed to be within [1, 2[ (i.e. an extracted mantissa)
Addiing exponent correction (optionnal) """
log2_hi_value = round(
log_f(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), RN)
log2_lo_value = round(
log_f(2) - log2_hi_value, self.precision.sollya_object, RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debug_multi)
rcp = ReciprocalSeed(_vx_mant, precision=self.precision, tag="rcp")
r = Multiplication(rcp, _vx_mant, precision=self.precision, tag="r")
int_format = self.precision.get_integer_format()
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=int_format),
Constant(-2, precision=int_format),
precision=int_format),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
C0 = Constant(0, precision=table_index.get_precision())
index_comp_0 = Equal(table_index,
C0,
tag="index_comp_0",
debug=debug_multi)
arg_red_index = Select(index_comp_0,
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_multi)
#_red_vx = arg_red_index * _vx_mant - 1.0
_red_vx = FMA(arg_red_index, _vx_mant, -1.0)
inv_err = S2**-6
red_interval = Interval(1 - inv_err, 1 + inv_err)
_red_vx.set_attributes(tag="_red_vx",
debug=debug_multi,
interval=red_interval)
# return in case of standard (non-special) input
if not tho_cond is None:
assert not log_table_tho is None
_log_inv_lo = Select(tho_cond,
TableLoad(log_table_tho, table_index, 1),
TableLoad(log_table, table_index, 1),
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = Select(tho_cond,
TableLoad(log_table_tho, table_index, 0),
TableLoad(log_table, table_index, 0),
tag="log_inv_hi",
debug=debug_multi)
else:
assert log_table_tho is None
_log_inv_lo = TableLoad(log_table, table_index, 1)
_log_inv_hi = TableLoad(log_table, table_index, 0)
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + x) / x, poly_degree, [self.precision] * (poly_degree + 1),
approx_interval, sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Info, "generating polynomial evaluation scheme")
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Info, "{}", poly_object.get_sollya_object())
# _poly approximates log10(1+r)/r
# _poly * red_vx approximates log10(x)
m0h, m0l = Mul211(_red_vx, _poly)
m0h, m0l = Add212(_red_vx, m0h, m0l)
m0h.set_attributes(tag="m0h", debug=debug_multi)
m0l.set_attributes(tag="m0l")
if not corr_exp is None:
l0_h = corr_exp * log2_hi
l0_l = corr_exp * log2_lo
l0_h.set_attributes(tag="l0_h")
l0_l.set_attributes(tag="l0_l")
rh, rl = Add222(l0_h, l0_l, m0h, m0l)
else:
# bypass exponent addition if no exponent correction is disabled
rh, rl = m0h, m0l
rh.set_attributes(tag="rh0", debug=debug_multi)
rl.set_attributes(tag="rl0", debug=debug_multi)
rh, rl = Add222(-_log_inv_hi, -_log_inv_lo, rh, rl)
rh.set_attributes(tag="rh", debug=debug_multi)
rl.set_attributes(tag="rl", debug=debug_multi)
# FIXME: log<self.basis>(vx) is computed as log(vx) / log(self.basis)
# which could be optimized for some value of self.basis (e.g. 2)
if sollya.log(self.basis) != 1.0:
lbh = self.precision.round_sollya_object(1 /
sollya.log(self.basis))
lbl = self.precision.round_sollya_object(1 /
sollya.log(self.basis) -
lbh)
rh, rl = Mul222(rh, rl, lbh, lbl)
return rh
else:
return rh
def generate_scalar_scheme(self, vx):
abs_vx = Abs(vx, precision=self.precision)
FCT_LIMIT = 1.0
one_limit = search_bound_threshold(sollya.erf, FCT_LIMIT, 1.0, 10.0,
self.precision)
one_limit_exp = int(sollya.floor(sollya.log2(one_limit)))
Log.report(Log.Debug, "erf(x) = 1.0 limit is {}, with exp={}",
one_limit, one_limit_exp)
upper_approx_bound = 10
# empiral numbers
eps_exp = {ML_Binary32: -3, ML_Binary64: -5}[self.precision]
eps = S2**eps_exp
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(0, eps)
# fonction to approximate is erf(x) / x
# it is an even function erf(x) / x = erf(-x) / (-x)
approx_fct = sollya.erf(sollya.x) - (sollya.x)
poly_degree = int(
sup(
guessdegree(approx_fct, approx_interval, S2**
-(self.precision.get_field_size() + 5)))) + 1
poly_degree_list = list(range(1, poly_degree, 2))
Log.report(Log.Debug, "poly_degree is {} and list {}", poly_degree,
poly_degree_list)
global_poly_object = Polynomial.build_from_approximation(
approx_fct, poly_degree_list,
[self.precision] * len(poly_degree_list), approx_interval,
sollya.relative)
Log.report(
Log.Debug, "inform is {}",
dirtyinfnorm(approx_fct - global_poly_object.get_sollya_object(),
approx_interval))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
ext_precision = {
ML_Binary32: ML_SingleSingle,
ML_Binary64: ML_DoubleDouble,
}[self.precision]
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, abs_vx, unified_precision=self.precision)
result = FMA(pre_poly, abs_vx, abs_vx)
result.set_attributes(tag="result", debug=debug_multi)
eps_target = S2**-(self.precision.get_field_size() + 5)
def offset_div_function(fct):
return lambda offset: fct(sollya.x + offset)
# empiral numbers
field_size = {ML_Binary32: 6, ML_Binary64: 8}[self.precision]
near_indexing = SubFPIndexing(eps_exp, 0, 6, self.precision)
near_approx = generic_poly_split(offset_div_function(sollya.erf),
near_indexing, eps_target,
self.precision, abs_vx)
near_approx.set_attributes(tag="near_approx", debug=debug_multi)
def offset_function(fct):
return lambda offset: fct(sollya.x + offset)
medium_indexing = SubFPIndexing(1, one_limit_exp, 7, self.precision)
medium_approx = generic_poly_split(offset_function(sollya.erf),
medium_indexing, eps_target,
self.precision, abs_vx)
medium_approx.set_attributes(tag="medium_approx", debug=debug_multi)
# approximation for positive values
scheme = ConditionBlock(
abs_vx < eps, Return(result),
ConditionBlock(
abs_vx < near_indexing.get_max_bound(), Return(near_approx),
ConditionBlock(abs_vx < medium_indexing.get_max_bound(),
Return(medium_approx),
Return(Constant(1.0,
precision=self.precision)))))
return scheme
def generate_scalar_scheme(self, vx, inline_select=False):
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# r_interval = Interval(0, 1.0)
index_size = 3
r_interval = Interval(-2**(-index_size), 2**-index_size)
local_ulp = sup(ulp(2**r_interval, self.precision))
Log.report(Log.Info, "ulp: ", local_ulp)
error_goal = S2**-1 * local_ulp
Log.report(Log.Info, "error goal: ", error_goal)
sollya_precision = {
ML_Binary32: sollya.binary32,
ML_Binary64: sollya.binary64
}[self.precision]
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
# Argument Reduction
# r = x - floor(x), r >= 0
vx_floor = Floor(vx,
precision=self.precision,
tag='vx_floor',
debug=debug_multi)
vx_int = Conversion(vx_floor,
precision=int_precision,
tag="vx_int",
debug=debug_multi)
vx_intf = vx_floor # Conversion(vx_int, precision = self.precision)
vx_r = vx - vx_intf
r_hi = NearestInteger(vx_r * 2**index_size,
precision=self.precision,
tag="r_hi",
debug=debug_multi)
# clamping r_hi_int within table-size to make sure
# it does not exceeds hi_part_table when used to index it
r_hi_int = Max(
Min(
Conversion(r_hi,
precision=int_precision,
tag="r_hi_int",
debug=debug_multi), 2**index_size + 1), 0)
r_lo = vx_r - r_hi * 2**-index_size
r_lo.set_attributes(tag="r_lo", debug=debug_multi)
vx_r.set_attributes(tag="vx_r", debug=debug_multi)
degree = sup(guessdegree(2**(sollya.x), r_interval, error_goal)) + 2
precision_list = [1] + [self.precision] * degree
exp_X = ExponentInsertion(vx_int,
tag="exp_X",
debug=debug_multi,
precision=self.precision)
#Polynomial Approx
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_object, poly_error = Polynomial.build_from_approximation_with_error(
2**(sollya.x) - 1, degree, precision_list, r_interval,
sollya.absolute)
Log.report(Log.Info, "Poly : %s" % poly_object)
Log.report(Log.Info, "poly_error : ", poly_error)
poly = polynomial_scheme_builder(poly_object.sub_poly(start_index=1),
r_lo,
unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
hi_part_table = ML_NewTable(dimensions=[2**index_size + 1],
storage_precision=self.precision,
tag=self.uniquify_name("exp2_table"),
const=True)
for i in range(2**index_size + 1):
input_value = i * 2**-index_size
tab_value = self.precision.round_sollya_object(
sollya.SollyaObject(2)**(input_value))
hi_part_table[i] = tab_value
hi_part_value = TableLoad(hi_part_table,
r_hi_int,
precision=self.precision,
tag="hi_part_value",
debug=debug_multi)
#Handling special cases
oflow_bound = Constant(self.precision.get_emax() + 1,
precision=self.precision)
subnormal_bound = self.precision.get_emin_subnormal()
uflow_bound = self.precision.get_emin_normal()
Log.report(Log.Info, "oflow : ", oflow_bound)
#print "uflow : ", uflow_bound
#print "sub : ", subnormal_bound
test_overflow = Comparison(vx,
oflow_bound,
specifier=Comparison.GreaterOrEqual)
test_overflow.set_attributes(tag="oflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_underflow = Comparison(vx, uflow_bound, specifier=Comparison.Less)
test_underflow.set_attributes(tag="uflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_subnormal = Comparison(vx,
subnormal_bound,
specifier=Comparison.Greater)
test_subnormal.set_attributes(tag="sub_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
subnormal_offset = -(uflow_bound - vx_int)
subnormal_offset.set_attributes(tag="offset", debug=debug_multi)
exp_offset = ExponentInsertion(subnormal_offset,
precision=self.precision,
debug=debug_multi,
tag="exp_offset")
exp_min = ExponentInsertion(uflow_bound,
precision=self.precision,
debug=debug_multi,
tag="exp_min")
subnormal_result = hi_part_value * exp_offset * exp_min * poly + hi_part_value * exp_offset * exp_min
test_std = LogicalOr(test_overflow,
test_underflow,
precision=ML_Bool,
tag="std_test",
likely=False,
debug=debug_multi)
#Reconstruction
result = hi_part_value * exp_X * poly + hi_part_value * exp_X
result.set_attributes(tag="result", debug=debug_multi)
C0 = Constant(0, precision=self.precision)
if inline_select:
scheme = Select(
test_std,
Select(test_overflow, FP_PlusInfty(self.precision),
Select(
test_subnormal,
subnormal_result,
C0,
)),
result,
)
return scheme
else:
return_inf = Return(FP_PlusInfty(self.precision))
return_C0 = Return(C0)
return_sub = Return(subnormal_result)
return_std = Return(result)
non_std_statement = Statement(
ConditionBlock(
test_overflow, return_inf,
ConditionBlock(test_subnormal, return_sub, return_C0)))
scheme = Statement(
ConditionBlock(test_std, non_std_statement, return_std))
return scheme
def generate_argument_reduction(self, memory_limit):
best_arg_reduc = None
best_arg_reduc = self.eval_argument_reduction(6,10,12,13)
best_arg_reduc['sizeof_tables'] = best_arg_reduc['sizeof_table1'] + best_arg_reduc['sizeof_table2']
best_arg_reduc['degree_poly1'] = 4
best_arg_reduc['degree_poly2'] = 8
return best_arg_reduc
# iterate through all possible parameters, and return the best argument reduction
# the order of importance of the caracteristics of a good argument reduction is:
# 1- the argument reduction is valid
# 2- the degree of the polynomials obtains are minimals
# 3- the memory used is minimal
# An arument reduction is valid iff:
# - the memory used is less than memory_limit
# - y-1 and z-1 fit into a uint64_t
# - the second argument reduction should usefull (ie: it should add at least 1 bit to the argument reduction)
# From thoses validity constraint we deduce some bound on the parameters to reduce the space of value searched:
# (note that thoses bound are implied by, but not equivalents to the constraints)
# size1 <= log2(memory_limit/17) (memory_limit on the first table)
# prec1 < 13 + size1 (y-1 fits into a uint64_t)
# size2 <= log2((memory_limit - sizeof_table1)/17/midinterval) (memory_limit on both tables)
# size2 >= 1 - log2(midinterval) (second arg red should be usefull)
# prec2 < 12 - prec1 - log2((y-y1)/y1), for all possible y (z-1 fits into a uint64_t)
# note: it is hard to deduce a tight bound on prec2 from the last inequality
# a good approximation is size2 ~= max[for y]( - log2((y-y1)/y1)), but using it may eliminate valid arg reduc
#self.eval_argument_reduction(12, 20, 22, 14)
min_size1 = 1
max_size1 = floor(log(memory_limit/17)/log(2)).getConstantAsInt()
for size1 in xrange(max_size1, min_size1-1, -1):
min_prec1 = size1
max_prec1 = 12 + size1
for prec1 in xrange(min_prec1,max_prec1+1):
# we need sizeof_table1 and mid_interval for the bound on size2 and prec2
first_arg_reduc = self.eval_argument_reduction(size1, prec1, prec1, prec1)
mid_interval = first_arg_reduc['mid_interval']
sizeof_table1 = first_arg_reduc['sizeof_table1']
if not(0 <= inf(mid_interval) and sup(mid_interval) < S2**(64 - 52 - prec1)):
continue
if not(first_arg_reduc['sizeof_table1'] < memory_limit):
continue
min_size2 = 1 - ceil(log(sup(mid_interval))/log(2)).getConstantAsInt()
max_size2 = floor(log((memory_limit - sizeof_table1)/(17 * sup(mid_interval)))/log(2)).getConstantAsInt()
# during execution of the prec2 loop, it can reduces the interval of valid values for prec2
# so min_prec2 and max_prec2 are setted here and not before the the prec2 loop
# (because they are modified inside the body of the loop, for the next iteration of size2)
min_prec2 = 0
max_prec2 = 12 + max_size2 - prec1
for size2 in xrange(max_size2,min_size2-1,-1):
max_prec2 = min(max_prec2, 12 + size2 - prec1)
for prec2 in xrange(max_prec2,min_prec2-1,-1):
#print '=====\t\033[1m{}\033[0m({}/{}),\t\033[1m{}\033[0m({}/{}),\t\033[1m{}\033[0m({}/{}),\t\033[1m{}\033[0m({}/{})\t====='.format(size1,min_size1,max_size1,prec1,min_prec1,max_prec1,size2,min_size2,max_size2,prec2,min_prec2,max_prec2)
#print resource.getrusage(resource.RUSAGE_SELF).ru_maxrss #memory used by the programm
arg_reduc = self.eval_argument_reduction(size1, prec1, size2, prec2)
mid_interval = arg_reduc['mid_interval']
out_interval = arg_reduc['out_interval']
sizeof_tables = arg_reduc['sizeof_table1'] + arg_reduc['sizeof_table2']
if not(0 <= inf(out_interval) and sup(out_interval) < S2**(64-52-prec1-prec2)):
max_prec2 = prec2 - 1
continue
if memory_limit < sizeof_tables:
continue
#assert(prec2 < 12 + size2 - prec1) # test the approximation size2 ~= max[for y]( - log2((y-y1)/y1))
# guess the degree of the two polynomials (relative error <= 2^-52 and absolute error <= 2^-120)
# note: we exclude zero from out_interval to not perturb sollya (log(1+x)/x is not well defined on 0)
sollya_out_interval = Interval(S2**(-52-prec1-prec2), sup(out_interval))
guess_degree_poly1 = guessdegree(log(1+sollya.x)/sollya.x, sollya_out_interval, S2**-52)
guess_degree_poly2 = guessdegree(log(1+sollya.x), sollya_out_interval, S2**-120)
# TODO: detect when guessdegree return multiple possible degree, and find the right one
if False and inf(guess_degree_poly1) <> sup(guess_degree_poly1):
print "improvable guess_degree_poly1:", guess_degree_poly1
if False and inf(guess_degree_poly2) <> sup(guess_degree_poly2):
print "improvable guess_degree_poly2:", guess_degree_poly2
degree_poly1 = sup(guess_degree_poly1).getConstantAsInt() + 1
degree_poly2 = sup(guess_degree_poly2).getConstantAsInt()
if ((best_arg_reduc is not None)
and (best_arg_reduc['degree_poly1'] < degree_poly1 or best_arg_reduc['degree_poly2'] < degree_poly2)):
min_prec2 = prec2 + 1
break
if ((best_arg_reduc is None)
or (best_arg_reduc['degree_poly1'] > degree_poly1)
or (best_arg_reduc['degree_poly1'] == degree_poly1 and best_arg_reduc['degree_poly2'] > degree_poly2)
or (best_arg_reduc['degree_poly1'] == degree_poly1 and best_arg_reduc['degree_poly2'] == degree_poly2 and best_arg_reduc['sizeof_tables'] > sizeof_tables)):
arg_reduc['degree_poly1'] = degree_poly1
arg_reduc['degree_poly2'] = degree_poly2
arg_reduc['sizeof_tables'] = sizeof_tables
best_arg_reduc = arg_reduc
#print "\n --new best-- \n", arg_reduc, "\n"
#print "\nBest arg reduc: \n", best_arg_reduc, "\n"
return best_arg_reduc
def generate_scalar_scheme(self, vx):
Log.set_dump_stdout(True)
Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
index_size = 5
comp_lo = (vx < 0)
comp_lo.set_attributes(tag = "comp_lo", precision = ML_Bool)
sign = Select(comp_lo, -1, 1, precision = self.precision)
# as sinh is an odd function, we can simplify the input to its absolute
# value once the sign has been extracted
vx = Abs(vx)
int_precision = self.precision.get_integer_format()
# argument reduction
arg_reg_value = log(2)/2**index_size
inv_log2_value = round(1/arg_reg_value, self.precision.get_sollya_object(), sollya.RN)
inv_log2_cst = Constant(inv_log2_value, precision = self.precision, tag = "inv_log2")
# for r_hi to be accurate we ensure k * log2_hi_value_cst is exact
# by limiting the number of non-zero bits in log2_hi_value_cst
# cosh(x) ~ exp(abs(x))/2 for a big enough x
# cosh(x) > 2^1023 <=> exp(x) > 2^1024 <=> x > log(2^1024)
# k = inv_log2_value * x
# -1 for guard
max_k_approx = inv_log2_value * log(sollya.SollyaObject(2)**1024)
max_k_bitsize = int(ceil(log2(max_k_approx)))
Log.report(Log.Info, "max_k_bitsize: %d" % max_k_bitsize)
log2_hi_value_precision = self.precision.get_precision() - max_k_bitsize - 1
log2_hi_value = round(arg_reg_value, log2_hi_value_precision, sollya.RN)
log2_lo_value = round(arg_reg_value - log2_hi_value, self.precision.get_sollya_object(), sollya.RN)
log2_hi_value_cst = Constant(log2_hi_value, tag = "log2_hi_value", precision = self.precision)
log2_lo_value_cst = Constant(log2_lo_value, tag = "log2_lo_value", precision = self.precision)
k = Trunc(Multiplication(inv_log2_cst, vx), precision = self.precision)
k_log2 = Multiplication(k, log2_hi_value_cst, precision = self.precision, exact = True, tag = "k_log2", unbreakable = True)
r_hi = vx - k_log2
r_hi.set_attributes(tag = "r_hi", debug = debug_multi, unbreakable = True)
r_lo = -k * log2_lo_value_cst
# reduced argument
r = r_hi + r_lo
r.set_attributes(tag = "r", debug = debug_multi)
if is_gappa_installed():
r_eval_error = self.get_eval_error(r_hi, variable_copy_map =
{
vx: Variable("vx", interval = Interval(0, 715), precision = self.precision),
k: Variable("k", interval = Interval(0, 1024), precision = self.precision)
})
Log.report(Log.Verbose, "r_eval_error: ", r_eval_error)
approx_interval = Interval(-arg_reg_value, arg_reg_value)
error_goal_approx = 2**-(self.precision.get_precision())
poly_degree = sup(guessdegree(exp(sollya.x), approx_interval, error_goal_approx)) + 3
precision_list = [1] + [self.precision] * (poly_degree)
k_integer = Conversion(k, precision = int_precision, tag = "k_integer", debug = debug_multi)
k_hi = BitLogicRightShift(k_integer, Constant(index_size, precision=int_precision), tag = "k_int_hi", precision = int_precision, debug = debug_multi)
k_lo = Modulo(k_integer, 2**index_size, tag = "k_int_lo", precision = int_precision, debug = debug_multi)
pow_exp = ExponentInsertion(Conversion(k_hi, precision = int_precision), precision = self.precision, tag = "pow_exp", debug = debug_multi)
exp_table = ML_NewTable(dimensions = [2 * 2**index_size, 4], storage_precision = self.precision, tag = self.uniquify_name("exp2_table"))
for i in range(2 * 2**index_size):
input_value = i - 2**index_size if i >= 2**index_size else i
reduced_hi_prec = int(self.precision.get_mantissa_size() - 8)
# using SollyaObject wrapper to force evaluation by sollya
# with higher precision
exp_value = sollya.SollyaObject(2)**((input_value)* 2**-index_size)
mexp_value = sollya.SollyaObject(2)**((-input_value)* 2**-index_size)
pos_value_hi = round(exp_value, reduced_hi_prec, sollya.RN)
pos_value_lo = round(exp_value - pos_value_hi, self.precision.get_sollya_object(), sollya.RN)
neg_value_hi = round(mexp_value, reduced_hi_prec, sollya.RN)
neg_value_lo = round(mexp_value - neg_value_hi, self.precision.get_sollya_object(), sollya.RN)
exp_table[i][0] = neg_value_hi
exp_table[i][1] = neg_value_lo
exp_table[i][2] = pos_value_hi
exp_table[i][3] = pos_value_lo
# log2_value = log(2) / 2^index_size
# sinh(x) = 1/2 * (exp(x) - exp(-x))
# exp(x) = exp(x - k * log2_value + k * log2_value)
#
# r = x - k * log2_value
# exp(x) = exp(r) * 2 ^ (k / 2^index_size)
#
# k / 2^index_size = h + l * 2^-index_size, with k, h, l integers
# exp(x) = exp(r) * 2^h * 2^(l *2^-index_size)
#
# sinh(x) = exp(r) * 2^(h-1) * 2^(l *2^-index_size) - exp(-r) * 2^(-h-1) * 2^(-l *2^-index_size)
# S=2^(h-1), T = 2^(-h-1)
# exp(r) = 1 + poly_pos(r)
# exp(-r) = 1 + poly_neg(r)
# 2^(l / 2^index_size) = pos_value_hi + pos_value_lo
# 2^(-l / 2^index_size) = neg_value_hi + neg_value_lo
#
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(exp(sollya.x), poly_degree, precision_list, approx_interval, sollya.absolute, error_function = error_function)
Log.report(Log.Verbose, "poly_approx_error: {}, {}".format(poly_approx_error, float(log2(poly_approx_error))))
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_pos = polynomial_scheme_builder(poly_object.sub_poly(start_index = 1), r, unified_precision = self.precision)
poly_pos.set_attributes(tag = "poly_pos", debug = debug_multi)
poly_neg = polynomial_scheme_builder(poly_object.sub_poly(start_index = 1), -r, unified_precision = self.precision)
poly_neg.set_attributes(tag = "poly_neg", debug = debug_multi)
table_index = Addition(k_lo, Constant(2**index_size, precision = int_precision), precision = int_precision, tag = "table_index", debug = debug_multi)
neg_value_load_hi = TableLoad(exp_table, table_index, 0, tag = "neg_value_load_hi", debug = debug_multi)
neg_value_load_lo = TableLoad(exp_table, table_index, 1, tag = "neg_value_load_lo", debug = debug_multi)
pos_value_load_hi = TableLoad(exp_table, table_index, 2, tag = "pos_value_load_hi", debug = debug_multi)
pos_value_load_lo = TableLoad(exp_table, table_index, 3, tag = "pos_value_load_lo", debug = debug_multi)
k_plus = Max(
Subtraction(k_hi, Constant(1, precision = int_precision), precision=int_precision, tag="k_plus", debug=debug_multi),
Constant(self.precision.get_emin_normal(), precision = int_precision))
k_neg = Max(
Subtraction(-k_hi, Constant(1, precision=int_precision), precision=int_precision, tag="k_neg", debug=debug_multi),
Constant(self.precision.get_emin_normal(), precision = int_precision))
# 2^(h-1)
pow_exp_pos = ExponentInsertion(k_plus, precision = self.precision, tag="pow_exp_pos", debug=debug_multi)
# 2^(-h-1)
pow_exp_neg = ExponentInsertion(k_neg, precision = self.precision, tag="pow_exp_neg", debug=debug_multi)
hi_terms = (pos_value_load_hi * pow_exp_pos - neg_value_load_hi * pow_exp_neg)
hi_terms.set_attributes(tag = "hi_terms", debug=debug_multi)
pos_exp = (pos_value_load_hi * poly_pos + (pos_value_load_lo + pos_value_load_lo * poly_pos)) * pow_exp_pos
pos_exp.set_attributes(tag = "pos_exp", debug = debug_multi)
neg_exp = (neg_value_load_hi * poly_neg + (neg_value_load_lo + neg_value_load_lo * poly_neg)) * pow_exp_neg
neg_exp.set_attributes(tag = "neg_exp", debug = debug_multi)
result = Addition(
Subtraction(
pos_exp,
neg_exp,
precision=self.precision,
),
hi_terms,
precision=self.precision,
tag="result",
debug=debug_multi
)
# ov_value
ov_value = round(asinh(self.precision.get_max_value()), self.precision.get_sollya_object(), sollya.RD)
ov_flag = Comparison(Abs(vx), Constant(ov_value, precision = self.precision), specifier = Comparison.Greater)
# main scheme
scheme = Statement(
Return(
Select(
ov_flag,
sign*FP_PlusInfty(self.precision),
sign*result
)))
return scheme
def generate_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = Abs(self.implementation.add_input_variable("x", self.precision),
tag="vx")
Log.report(Log.Info, "generating implementation scheme")
if self.debug_flag:
Log.report(Log.Info, "debug has been enabled")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
debug_precision = {
ML_Binary32: debug_ftox,
ML_Binary64: debug_lftolx
}[self.precision]
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
# return in case of infinity input
infty_return = Statement(
ConditionBlock(test_positive, Return(FP_PlusInfty(self.precision)),
Return(FP_PlusZero(self.precision))))
# return in case of specific value input (NaN or inf)
specific_return = ConditionBlock(
test_nan,
ConditionBlock(test_signaling_nan, return_snan,
Return(FP_QNaN(self.precision))), infty_return)
# return in case of standard (non-special) input
sollya_precision = self.precision.get_sollya_object()
hi_precision = self.precision.get_field_size() - 3
# argument reduction
frac_pi_index = 3
frac_pi = round(S2**frac_pi_index / pi, sollya_precision, sollya.RN)
inv_frac_pi = round(pi / S2**frac_pi_index, hi_precision, sollya.RN)
inv_frac_pi_lo = round(pi / S2**frac_pi_index - inv_frac_pi,
sollya_precision, sollya.RN)
# computing k = E(x * frac_pi)
vx_pi = Multiplication(vx, frac_pi, precision=self.precision)
k = NearestInteger(vx_pi, precision=ML_Int32, tag="k", debug=True)
fk = Conversion(k, precision=self.precision, tag="fk")
inv_frac_pi_cst = Constant(inv_frac_pi,
tag="inv_frac_pi",
precision=self.precision)
inv_frac_pi_lo_cst = Constant(inv_frac_pi_lo,
tag="inv_frac_pi_lo",
precision=self.precision)
red_vx_hi = (vx - inv_frac_pi_cst * fk)
red_vx_hi.set_attributes(tag="red_vx_hi",
debug=debug_precision,
precision=self.precision)
red_vx_lo_sub = inv_frac_pi_lo_cst * fk
red_vx_lo_sub.set_attributes(tag="red_vx_lo_sub",
debug=debug_precision,
unbreakable=True,
precision=self.precision)
vx_d = Conversion(vx, precision=ML_Binary64, tag="vx_d")
pre_red_vx = red_vx_hi - inv_frac_pi_lo_cst * fk
pre_red_vx_d_hi = (vx_d - inv_frac_pi_cst * fk)
pre_red_vx_d_hi.set_attributes(tag="pre_red_vx_d_hi",
precision=ML_Binary64,
debug=debug_lftolx)
pre_red_vx_d = pre_red_vx_d_hi - inv_frac_pi_lo_cst * fk
pre_red_vx_d.set_attributes(tag="pre_red_vx_d",
debug=debug_lftolx,
precision=ML_Binary64)
modk = Modulo(k,
2**(frac_pi_index + 1),
precision=ML_Int32,
tag="switch_value",
debug=True)
sel_c = Equal(BitLogicAnd(modk, 2**(frac_pi_index - 1)),
2**(frac_pi_index - 1))
red_vx = Select(sel_c, -pre_red_vx, pre_red_vx)
red_vx.set_attributes(tag="red_vx",
debug=debug_precision,
precision=self.precision)
red_vx_d = Select(sel_c, -pre_red_vx_d, pre_red_vx_d)
red_vx_d.set_attributes(tag="red_vx_d",
debug=debug_lftolx,
precision=ML_Binary64)
approx_interval = Interval(-pi / (S2**(frac_pi_index + 1)),
pi / S2**(frac_pi_index + 1))
Log.report(Log.Info, "approx interval: %s\n" % approx_interval)
error_goal_approx = S2**-self.precision.get_precision()
Log.report(Log.Info, "building mathematical polynomial")
poly_degree_vector = [None] * 2**(frac_pi_index + 1)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
#polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
index_relative = []
poly_object_vector = [None] * 2**(frac_pi_index + 1)
for i in range(2**(frac_pi_index + 1)):
sub_func = cos(sollya.x + i * pi / S2**frac_pi_index)
degree = int(
sup(guessdegree(sub_func, approx_interval,
error_goal_approx))) + 1
degree_list = range(degree + 1)
a_interval = approx_interval
if i == 0:
# ad-hoc, TODO: to be cleaned
degree = 6
degree_list = range(0, degree + 1, 2)
elif i % 2**(frac_pi_index) == 2**(frac_pi_index - 1):
# for pi/2 and 3pi/2, an approx to sin=cos(pi/2+x)
# must be generated
degree_list = range(1, degree + 1, 2)
if i == 3 or i == 5 or i == 7 or i == 9:
precision_list = [sollya.binary64
] + [sollya.binary32] * (degree)
else:
precision_list = [sollya.binary32] * (degree + 1)
poly_degree_vector[i] = degree
constraint = sollya.absolute
delta = (2**(frac_pi_index - 3))
centered_i = (i % 2**(frac_pi_index)) - 2**(frac_pi_index - 1)
if centered_i < delta and centered_i > -delta and centered_i != 0:
constraint = sollya.relative
index_relative.append(i)
Log.report(
Log.Info, "generating approximation for %d/%d" %
(i, 2**(frac_pi_index + 1)))
poly_object_vector[
i], _ = Polynomial.build_from_approximation_with_error(
sub_func,
degree_list,
precision_list,
a_interval,
constraint,
error_function=error_function)
# unified power map for red_sx^n
upm = {}
rel_error_list = []
poly_scheme_vector = [None] * (2**(frac_pi_index + 1))
for i in range(2**(frac_pi_index + 1)):
poly_object = poly_object_vector[i]
poly_precision = self.precision
if i == 3 or i == 5 or i == 7 or i == 9:
poly_precision = ML_Binary64
c0 = Constant(coeff(poly_object.get_sollya_object(), 0),
precision=ML_Binary64)
c1 = Constant(coeff(poly_object.get_sollya_object(), 1),
precision=self.precision)
poly_hi = (c0 + c1 * red_vx)
poly_hi.set_precision(ML_Binary64)
red_vx_d_2 = red_vx_d * red_vx_d
poly_scheme = poly_hi + red_vx_d_2 * polynomial_scheme_builder(
poly_object.sub_poly(start_index=2, offset=2),
red_vx,
unified_precision=self.precision,
power_map_=upm)
poly_scheme.set_attributes(unbreakable=True)
elif i == 4:
c1 = Constant(coeff(poly_object.get_sollya_object(), 1),
precision=ML_Binary64)
poly_scheme = c1 * red_vx_d + polynomial_scheme_builder(
poly_object.sub_poly(start_index=2),
red_vx,
unified_precision=self.precision,
power_map_=upm)
poly_scheme.set_precision(ML_Binary64)
else:
poly_scheme = polynomial_scheme_builder(
poly_object,
red_vx,
unified_precision=poly_precision,
power_map_=upm)
#if i == 3:
# c0 = Constant(coeff(poly_object.get_sollya_object(), 0), precision = self.precision)
# c1 = Constant(coeff(poly_object.get_sollya_object(), 1), precision = self.precision)
# poly_scheme = (c0 + c1 * red_vx) + polynomial_scheme_builder(poly_object.sub_poly(start_index = 2), red_vx, unified_precision = self.precision, power_map_ = upm)
poly_scheme.set_attributes(tag="poly_cos%dpi%d" %
(i, 2**(frac_pi_index)),
debug=debug_precision)
poly_scheme_vector[i] = poly_scheme
#try:
if is_gappa_installed() and i == 3:
opt_scheme = self.opt_engine.optimization_process(
poly_scheme,
self.precision,
copy=True,
fuse_fma=self.fuse_fma)
tag_map = {}
self.opt_engine.register_nodes_by_tag(opt_scheme, tag_map)
gappa_vx = Variable("red_vx",
precision=self.precision,
interval=approx_interval)
cg_eval_error_copy_map = {
tag_map["red_vx"]: gappa_vx,
tag_map["red_vx_d"]: gappa_vx,
}
print "opt_scheme"
print opt_scheme.get_str(depth=None,
display_precision=True,
memoization_map={})
eval_error = self.gappa_engine.get_eval_error_v2(
self.opt_engine,
opt_scheme,
cg_eval_error_copy_map,
gappa_filename="red_arg_%d.g" % i)
poly_range = cos(approx_interval + i * pi / S2**frac_pi_index)
rel_error_list.append(eval_error / poly_range)
#for rel_error in rel_error_list:
# print sup(abs(rel_error))
#return
# case 17
#poly17 = poly_object_vector[17]
#c0 = Constant(coeff(poly17.get_sollya_object(), 0), precision = self.precision)
#c1 = Constant(coeff(poly17.get_sollya_object(), 1), precision = self.precision)
#poly_scheme_vector[17] = FusedMultiplyAdd(c1, red_vx, c0, specifier = FusedMultiplyAdd.Standard) + polynomial_scheme_builder(poly17.sub_poly(start_index = 2), red_vx, unified_precision = self.precision, power_map_ = upm)
half = 2**frac_pi_index
sub_half = 2**(frac_pi_index - 1)
# determine if the reduced input is within the second and third quarter (not first nor fourth)
# to negate the cosine output
factor_cond = BitLogicAnd(BitLogicXor(
BitLogicRightShift(modk, frac_pi_index),
BitLogicRightShift(modk, frac_pi_index - 1)),
1,
tag="factor_cond",
debug=True)
CM1 = Constant(-1, precision=self.precision)
C1 = Constant(1, precision=self.precision)
factor = Select(factor_cond,
CM1,
C1,
tag="factor",
debug=debug_precision)
factor2 = Select(Equal(modk, Constant(sub_half)),
CM1,
C1,
tag="factor2",
debug=debug_precision)
switch_map = {}
if 0:
for i in range(2**(frac_pi_index + 1)):
switch_map[i] = Return(poly_scheme_vector[i])
else:
for i in range(2**(frac_pi_index - 1)):
switch_case = (i, half - i)
#switch_map[i] = Return(poly_scheme_vector[i])
#switch_map[half-i] = Return(-poly_scheme_vector[i])
if i != 0:
switch_case = switch_case + (half + i, 2 * half - i)
#switch_map[half+i] = Return(-poly_scheme_vector[i])
#switch_map[2*half-i] = Return(poly_scheme_vector[i])
if poly_scheme_vector[i].get_precision() != self.precision:
poly_result = Conversion(poly_scheme_vector[i],
precision=self.precision)
else:
poly_result = poly_scheme_vector[i]
switch_map[switch_case] = Return(factor * poly_result)
#switch_map[sub_half] = Return(-poly_scheme_vector[sub_half])
#switch_map[half + sub_half] = Return(poly_scheme_vector[sub_half])
switch_map[(sub_half, half + sub_half)] = Return(
factor2 * poly_scheme_vector[sub_half])
result = SwitchBlock(modk, switch_map)
#######################################################################
# LARGE ARGUMENT MANAGEMENT #
# (lar: Large Argument Reduction) #
#######################################################################
# payne and hanek argument reduction for large arguments
#red_func_name = "payne_hanek_cosfp32" # "payne_hanek_fp32_asm"
red_func_name = "payne_hanek_fp32_asm"
payne_hanek_func_op = FunctionOperator(
red_func_name,
arg_map={0: FO_Arg(0)},
require_header=["support_lib/ml_red_arg.h"])
payne_hanek_func = FunctionObject(red_func_name, [ML_Binary32],
ML_Binary64, payne_hanek_func_op)
payne_hanek_func_op.declare_prototype = payne_hanek_func
#large_arg_red = FunctionCall(payne_hanek_func, vx)
large_arg_red = payne_hanek_func(vx)
red_bound = S2**20
cond = Abs(vx) >= red_bound
cond.set_attributes(tag="cond", likely=False)
lar_neark = NearestInteger(large_arg_red, precision=ML_Int64)
lar_modk = Modulo(lar_neark,
Constant(16, precision=ML_Int64),
tag="lar_modk",
debug=True)
# Modulo is supposed to be already performed (by payne_hanek_cosfp32)
#lar_modk = NearestInteger(large_arg_red, precision = ML_Int64)
pre_lar_red_vx = large_arg_red - Conversion(lar_neark,
precision=ML_Binary64)
pre_lar_red_vx.set_attributes(precision=ML_Binary64,
debug=debug_lftolx,
tag="pre_lar_red_vx")
lar_red_vx = Conversion(pre_lar_red_vx,
precision=self.precision,
debug=debug_precision,
tag="lar_red_vx")
lar_red_vx_lo = Conversion(
pre_lar_red_vx - Conversion(lar_red_vx, precision=ML_Binary64),
precision=self.precision)
lar_red_vx_lo.set_attributes(tag="lar_red_vx_lo",
precision=self.precision)
lar_k = 3
# large arg reduction Universal Power Map
lar_upm = {}
lar_switch_map = {}
approx_interval = Interval(-0.5, 0.5)
for i in range(2**(lar_k + 1)):
frac_pi = pi / S2**lar_k
func = cos(frac_pi * i + frac_pi * sollya.x)
degree = 6
error_mode = sollya.absolute
if i % 2**(lar_k) == 2**(lar_k - 1):
# close to sin(x) cases
func = -sin(frac_pi * x) if i == 2**(lar_k -
1) else sin(frac_pi * x)
degree_list = range(0, degree + 1, 2)
precision_list = [sollya.binary32] * len(degree_list)
poly_object, _ = Polynomial.build_from_approximation_with_error(
func / x, degree_list, precision_list, approx_interval,
error_mode)
poly_object = poly_object.sub_poly(offset=-1)
else:
degree_list = range(degree + 1)
precision_list = [sollya.binary32] * len(degree_list)
poly_object, _ = Polynomial.build_from_approximation_with_error(
func, degree_list, precision_list, approx_interval,
error_mode)
if i == 3 or i == 5 or i == 7 or i == 9 or i == 11 or i == 13:
poly_precision = ML_Binary64
c0 = Constant(coeff(poly_object.get_sollya_object(), 0),
precision=ML_Binary64)
c1 = Constant(coeff(poly_object.get_sollya_object(), 1),
precision=self.precision)
poly_hi = (c0 + c1 * lar_red_vx)
poly_hi.set_precision(ML_Binary64)
pre_poly_scheme = poly_hi + polynomial_scheme_builder(
poly_object.sub_poly(start_index=2),
lar_red_vx,
unified_precision=self.precision,
power_map_=lar_upm)
pre_poly_scheme.set_attributes(precision=ML_Binary64)
poly_scheme = Conversion(pre_poly_scheme,
precision=self.precision)
elif i == 4 or i == 12:
c1 = Constant(coeff(poly_object.get_sollya_object(), 1),
precision=self.precision)
c3 = Constant(coeff(poly_object.get_sollya_object(), 3),
precision=self.precision)
c5 = Constant(coeff(poly_object.get_sollya_object(), 5),
precision=self.precision)
poly_hi = polynomial_scheme_builder(
poly_object.sub_poly(start_index=3),
lar_red_vx,
unified_precision=self.precision,
power_map_=lar_upm)
poly_hi.set_attributes(tag="poly_lar_%d_hi" % i,
precision=ML_Binary64)
poly_scheme = Conversion(FusedMultiplyAdd(
c1, lar_red_vx, poly_hi, precision=ML_Binary64) +
c1 * lar_red_vx_lo,
precision=self.precision)
else:
poly_scheme = polynomial_scheme_builder(
poly_object,
lar_red_vx,
unified_precision=self.precision,
power_map_=lar_upm)
# poly_scheme = polynomial_scheme_builder(poly_object, lar_red_vx, unified_precision = self.precision, power_map_ = lar_upm)
poly_scheme.set_attributes(tag="lar_poly_%d" % i,
debug=debug_precision)
lar_switch_map[(i, )] = Return(poly_scheme)
lar_result = SwitchBlock(lar_modk, lar_switch_map)
# main scheme
#Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
# scheme = Statement(ConditionBlock(cond, lar_result, result))
Log.report(Log.Info, "Construction of the initial MDL scheme")
scheme = Statement(pre_red_vx_d, red_vx_lo_sub,
ConditionBlock(cond, lar_result, result))
return scheme
def generate_scalar_scheme(self, vx):
# approximation the gamma function
abs_vx = Abs(vx, precision=self.precision)
FCT_LIMIT = 1.0
omega_value = self.precision.get_omega()
def sollya_wrap_bigfloat_fct(bfct):
""" wrap bigfloat's function <bfct> such that is can be used
on SollyaObject inputs and returns SollyaObject results """
def fct(x):
return sollya.SollyaObject(bfct(SollyaObject(x).bigfloat()))
return fct
sollya_gamma = sollya_wrap_bigfloat_fct(bigfloat.gamma)
sollya_digamma = sollya_wrap_bigfloat_fct(bigfloat.digamma)
# first derivative of gamma is digamma * gamma
bigfloat_gamma_d0 = lambda x: bigfloat.gamma(x) * bigfloat.digamma(x)
sollya_gamma_d0 = sollya_wrap_bigfloat_fct(bigfloat_gamma_d0)
# approximating trigamma with straightforward derivatives formulae of digamma
U = 2**-64
bigfloat_trigamma = lambda x: (
(bigfloat.digamma(x * (1 + U)) - bigfloat.digamma(x)) / (x * U))
sollya_trigamma = sollya_wrap_bigfloat_fct(bigfloat_trigamma)
bigfloat_gamma_d1 = lambda x: (bigfloat_trigamma(x) * bigfloat.gamma(
x) + bigfloat_gamma_d0(x) * bigfloat.digamma(x))
sollya_gamma_d1 = sollya_wrap_bigfloat_fct(bigfloat_gamma_d1)
def sollya_gamma_fct(x, diff_order, prec):
""" wrapper to use bigfloat implementation of exponential
rather than sollya's implementation directly.
This wrapper implements sollya's function API.
:param x: numerical input value (may be an Interval)
:param diff_order: differential order
:param prec: numerical precision expected (min)
"""
fct = None
if diff_order == 0:
fct = sollya_gamma
elif diff_order == 1:
fct = sollya_gamma_d0
elif diff_order == 2:
fct = sollya_gamma_d1
else:
raise NotImplementedError
with bigfloat.precision(prec):
if x.is_range():
lo = sollya.inf(x)
hi = sollya.sup(x)
return sollya.Interval(fct(lo), fct(hi))
else:
return fct(x)
# search the lower x such that gamma(x) >= omega
omega_upper_limit = search_bound_threshold(sollya_gamma, omega_value,
2, 1000.0, self.precision)
Log.report(Log.Debug, "gamma(x) = {} limit is {}", omega_value,
omega_upper_limit)
# evaluate gamma(<min-normal-value>)
lower_x_bound = self.precision.get_min_normal_value()
value_min = sollya_gamma(lower_x_bound)
Log.report(Log.Debug, "gamma({}) = {}(log2={})", lower_x_bound,
value_min, int(sollya.log2(value_min)))
# evaluate gamma(<min-subnormal-value>)
lower_x_bound = self.precision.get_min_subnormal_value()
value_min = sollya_gamma(lower_x_bound)
Log.report(Log.Debug, "gamma({}) = {}(log2={})", lower_x_bound,
value_min, int(sollya.log2(value_min)))
# Gamma is defined such that gamma(x+1) = x * gamma(x)
#
# we approximate gamma over [1, 2]
# y in [1, 2]
# gamma(y) = (y-1) * gamma(y-1)
# gamma(y-1) = gamma(y) / (y-1)
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(1, 2)
approx_fct = sollya.function(sollya_gamma_fct)
poly_degree = int(
sup(
guessdegree(approx_fct, approx_interval, S2**
-(self.precision.get_field_size() + 5)))) + 1
Log.report(Log.Debug, "approximation's poly degree over [1, 2] is {}",
poly_degree)
sys.exit(1)
poly_degree_list = list(range(1, poly_degree, 2))
Log.report(Log.Debug, "poly_degree is {} and list {}", poly_degree,
poly_degree_list)
global_poly_object = Polynomial.build_from_approximation(
approx_fct, poly_degree_list,
[self.precision] * len(poly_degree_list), approx_interval,
sollya.relative)
Log.report(
Log.Debug, "inform is {}",
dirtyinfnorm(approx_fct - global_poly_object.get_sollya_object(),
approx_interval))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
ext_precision = {
ML_Binary32: ML_SingleSingle,
ML_Binary64: ML_DoubleDouble,
}[self.precision]
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, abs_vx, unified_precision=self.precision)
result = FMA(pre_poly, abs_vx, abs_vx)
result.set_attributes(tag="result", debug=debug_multi)
eps_target = S2**-(self.precision.get_field_size() + 5)
def offset_div_function(fct):
return lambda offset: fct(sollya.x + offset)
# empiral numbers
field_size = {ML_Binary32: 6, ML_Binary64: 8}[self.precision]
near_indexing = SubFPIndexing(eps_exp, 0, 6, self.precision)
near_approx = generic_poly_split(offset_div_function(sollya.erf),
near_indexing, eps_target,
self.precision, abs_vx)
near_approx.set_attributes(tag="near_approx", debug=debug_multi)
def offset_function(fct):
return lambda offset: fct(sollya.x + offset)
medium_indexing = SubFPIndexing(1, one_limit_exp, 7, self.precision)
medium_approx = generic_poly_split(offset_function(sollya.erf),
medium_indexing, eps_target,
self.precision, abs_vx)
medium_approx.set_attributes(tag="medium_approx", debug=debug_multi)
# approximation for positive values
scheme = ConditionBlock(
abs_vx < eps, Return(result),
ConditionBlock(
abs_vx < near_indexing.get_max_bound(), Return(near_approx),
ConditionBlock(abs_vx < medium_indexing.get_max_bound(),
Return(medium_approx),
Return(Constant(1.0,
precision=self.precision)))))
return scheme
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
r_interval = Interval(-0.5, 0.5)
local_ulp = sup(ulp(2**r_interval, self.precision))
Log.report(Log.Info, "ulp: ", local_ulp)
error_goal = S2**-1 * local_ulp
Log.report(Log.Info, "error goal: ", error_goal)
sollya_precision = {
ML_Binary32: sollya.binary32,
ML_Binary64: sollya.binary64
}[self.precision]
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
#Argument Reduction
vx_int = NearestInteger(vx,
precision=int_precision,
tag='vx_int',
debug=debug_multi)
vx_intf = Conversion(vx_int, precision=self.precision)
vx_r = vx - vx_intf
vx_r.set_attributes(tag="vx_r", debug=debug_multi)
degree = sup(guessdegree(2**(sollya.x), r_interval, error_goal)) + 2
precision_list = [1] + [self.precision] * degree
exp_X = ExponentInsertion(vx_int,
tag="exp_X",
debug=debug_multi,
precision=self.precision)
#Polynomial Approx
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_object, poly_error = Polynomial.build_from_approximation_with_error(
2**(sollya.x) - 1, degree, precision_list, r_interval,
sollya.absolute)
Log.report(Log.Info, "Poly : %s" % poly_object)
Log.report(Log.Info, "poly_error : ", poly_error)
poly = polynomial_scheme_builder(poly_object.sub_poly(start_index=1),
vx_r,
unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
#Handling special cases
oflow_bound = Constant(self.precision.get_emax() + 1,
precision=self.precision)
subnormal_bound = self.precision.get_emin_subnormal()
uflow_bound = self.precision.get_emin_normal()
Log.report(Log.Info, "oflow : ", oflow_bound)
#print "uflow : ", uflow_bound
#print "sub : ", subnormal_bound
test_overflow = Comparison(vx,
oflow_bound,
specifier=Comparison.GreaterOrEqual)
test_overflow.set_attributes(tag="oflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_underflow = Comparison(vx, uflow_bound, specifier=Comparison.Less)
test_underflow.set_attributes(tag="uflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_subnormal = Comparison(vx,
subnormal_bound,
specifier=Comparison.Greater)
test_subnormal.set_attributes(tag="sub_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
subnormal_offset = -(uflow_bound - vx_int)
subnormal_offset.set_attributes(tag="offset", debug=debug_multi)
exp_offset = ExponentInsertion(subnormal_offset,
precision=self.precision,
debug=debug_multi,
tag="exp_offset")
exp_min = ExponentInsertion(uflow_bound,
precision=self.precision,
debug=debug_multi,
tag="exp_min")
subnormal_result = exp_offset * exp_min * poly + exp_offset * exp_min
test_std = LogicalOr(test_overflow,
test_underflow,
precision=ML_Bool,
tag="std_test",
likely=False)
#Reconstruction
result = exp_X * poly + exp_X
result.set_attributes(tag="result", debug=debug_multi)
C0 = Constant(0, precision=self.precision)
return_inf = Return(FP_PlusInfty(self.precision))
return_C0 = Return(C0)
return_sub = Return(subnormal_result)
return_std = Return(result)
non_std_statement = Statement(
ConditionBlock(
test_overflow, return_inf,
ConditionBlock(test_subnormal, return_sub, return_C0)))
scheme = Statement(
ConditionBlock(test_std, non_std_statement, return_std))
return scheme
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
debug=debug_multi,
precision=self.precision)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debug_multi)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(_vx_mant, precision=int_precision, debug=debug_multi),
self.precision.get_field_size() - 7,
debug=debug_multi),
0x7f,
tag="table_index",
debug=debug_multi)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=integer_precision),
Constant(-2, precision=integer_precision),
precision=integer_precision),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
arg_red_index = Select(Equal(table_index, 0), 1.0,
pre_arg_red_index)
#_red_vx = arg_red_index * _vx_mant - 1.0
_red_vx = FusedMultiplyAdd(arg_red_index,
_vx_mant,
1.0,
specifier=FusedMultiplyAdd.Subtract)
_red_vx.set_attributes(tag="_red_vx", debug=debug_multi)
inv_err = S2**-7
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_multi)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree,
[1] + [self.precision] * (poly_degree), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
_poly = PolynomialSchemeEvaluator.generate_estrin_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
corr_exp = Conversion(
_vx_exp if exp_corr_factor == None else _vx_exp +
exp_corr_factor,
precision=self.precision)
split_red_vx = Split(_red_vx,
precision=ML_DoubleDouble,
tag="split_red_vx",
debug=debug_multi)
red_vx_hi = split_red_vx.hi
red_vx_lo = split_red_vx.lo
# result = _red_vx * poly - log_inv_hi - log_inv_lo + _vx_exp * log2_hi + _vx_exp * log2_lo
pre_result = -_log_inv_hi + (_red_vx +
(_red_vx * _poly +
(corr_exp * log2_lo - _log_inv_lo)))
pre_result.set_attributes(tag="pre_result", debug=debug_multi)
exact_log2_hi_exp = corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_exp",
debug=debug_multi)
cancel_part = (corr_exp * log2_hi - _log_inv_hi)
cancel_part.set_attributes(tag="cancel_part", debug=debug_multi)
sub_part = red_vx_hi + cancel_part
sub_part.set_attributes(tag="sub_part", debug=debug_multi)
#result_one_low_part = (red_vx_hi * _poly + (red_vx_lo + (red_vx_lo * _poly + (corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part = ((red_vx_lo +
(red_vx_lo * _poly +
(corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part.set_attributes(tag="result_one_low_part",
debug=debug_multi)
_result_one = (
(sub_part) + red_vx_hi * _poly) + result_one_low_part
return exact_log2_hi_exp + pre_result, _poly, _log_inv_lo, _log_inv_hi, _red_vx, _result_one
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.precision.sollya_object
# constant computation
invlog2 = round(1 / log(2), sollya_precision, sollya.RN)
invlog2_cst = Constant(invlog2, precision=self.precision)
#v_log2_hi = round(log(2), 16, sollya.RN)
#v_log2_lo = round(log(2) - v_log2_hi, sollya_precision, sollya.RN)
#log2_hi = Constant(v_log2_hi, precision = self.precision, tag = "log2_hi")
#log2_lo = Constant(v_log2_lo, precision = self.precision, tag = "log2_lo")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
v_log2_hi = round(
log(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
v_log2_lo = round(
log(2) - v_log2_hi, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(v_log2_hi, precision=self.precision, tag="log2_hi")
log2_lo = Constant(v_log2_lo, precision=self.precision, tag="log2_lo")
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debug_multi)
int_precision = self.precision.get_integer_format()
# table creation
table_index_size = 7
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("inv_table"))
log_table[0][0] = 0.0
log_table[0][1] = 0.0
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
integer_precision = {
ML_Binary32: ML_UInt32,
ML_Binary64: ML_UInt64
}[self.precision]
for i in range(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
inv_value = inv_approx_table[
i] # (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
value_high = round(
log(inv_value),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(
log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
debug=debug_multi,
precision=self.precision)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debug_multi)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(_vx_mant, precision=int_precision, debug=debug_multi),
self.precision.get_field_size() - 7,
debug=debug_multi),
0x7f,
tag="table_index",
debug=debug_multi)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=integer_precision),
Constant(-2, precision=integer_precision),
precision=integer_precision),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
arg_red_index = Select(Equal(table_index, 0), 1.0,
pre_arg_red_index)
#_red_vx = arg_red_index * _vx_mant - 1.0
_red_vx = FusedMultiplyAdd(arg_red_index,
_vx_mant,
1.0,
specifier=FusedMultiplyAdd.Subtract)
_red_vx.set_attributes(tag="_red_vx", debug=debug_multi)
inv_err = S2**-7
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_multi)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree,
[1] + [self.precision] * (poly_degree), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
_poly = PolynomialSchemeEvaluator.generate_estrin_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
corr_exp = Conversion(
_vx_exp if exp_corr_factor == None else _vx_exp +
exp_corr_factor,
precision=self.precision)
split_red_vx = Split(_red_vx,
precision=ML_DoubleDouble,
tag="split_red_vx",
debug=debug_multi)
red_vx_hi = split_red_vx.hi
red_vx_lo = split_red_vx.lo
# result = _red_vx * poly - log_inv_hi - log_inv_lo + _vx_exp * log2_hi + _vx_exp * log2_lo
pre_result = -_log_inv_hi + (_red_vx +
(_red_vx * _poly +
(corr_exp * log2_lo - _log_inv_lo)))
pre_result.set_attributes(tag="pre_result", debug=debug_multi)
exact_log2_hi_exp = corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_exp",
debug=debug_multi)
cancel_part = (corr_exp * log2_hi - _log_inv_hi)
cancel_part.set_attributes(tag="cancel_part", debug=debug_multi)
sub_part = red_vx_hi + cancel_part
sub_part.set_attributes(tag="sub_part", debug=debug_multi)
#result_one_low_part = (red_vx_hi * _poly + (red_vx_lo + (red_vx_lo * _poly + (corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part = ((red_vx_lo +
(red_vx_lo * _poly +
(corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part.set_attributes(tag="result_one_low_part",
debug=debug_multi)
_result_one = (
(sub_part) + red_vx_hi * _poly) + result_one_low_part
return exact_log2_hi_exp + pre_result, _poly, _log_inv_lo, _log_inv_hi, _red_vx, _result_one
result, poly, log_inv_lo, log_inv_hi, red_vx, new_result_one = compute_log(
vx)
result.set_attributes(tag="result", debug=debug_multi)
new_result_one.set_attributes(tag="new_result_one", debug=debug_multi)
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debug_multi,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debug_multi,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debug_multi,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debug_multi,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debug_multi,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debug_multi,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debug_multi)
# exp=-1 case
Log.report(Log.Verbose, "managing exp=-1 case")
result2 = (-log_inv_hi - log2_hi) + (
(red_vx + poly * red_vx) - log2_lo - log_inv_lo)
result2.set_attributes(tag="result2", debug=debug_multi)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
Log.report(Log.Verbose, "managing close to 1.0 cases")
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
sollya.absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_multi)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one",
debug=debug_multi,
likely=False)
# main scheme
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
), Return(result_subnormal)),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result2),
Return(result))
#ConditionBlock(cond_one,
#Return(new_result_one),
#ConditionBlock(exp_mone,
#Return(result2),
#Return(result)
#)
#)
))))
scheme = pre_scheme
return scheme
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_lftolx)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debugd)
# The main table is indexed by the 7 most significant bits
# of the mantissa
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debuglld)
# argument reduction
# Using AND -2 to exclude LSB set to 1 for Newton-Raphson convergence
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(DivisionSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_lftolx,
silent=True),
precision=ML_UInt64),
Constant(-2, precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
_red_vx = FMA(arg_red_index, _vx_mant, -1.0)
_red_vx.set_attributes(tag="_red_vx", debug=debug_lftolx)
inv_err = S2**-inv_approx_table.index_size
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log2(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() * 1.1))) + 1
sollya.settings.display = sollya.hexadecimal
global_poly_object, approx_error = Polynomial.build_from_approximation_with_error(
log2(1 + sollya.x) / sollya.x,
poly_degree, [self.precision] * (poly_degree + 1),
approx_interval,
sollya.absolute,
error_function=lambda p, f, ai, mod, t: sollya.dirtyinfnorm(
p - f, ai))
Log.report(
Log.Info, "poly_degree={}, approx_error={}".format(
poly_degree, approx_error))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
#poly_object = global_poly_object.sub_poly(start_index=0,offset=0)
Attributes.set_default_silent(True)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly = FMA(pre_poly, _red_vx,
global_poly_object.get_cst_coeff(0, self.precision))
_poly.set_attributes(tag="poly", debug=debug_lftolx)
Log.report(
Log.Verbose, "sollya global_poly_object: {}".format(
global_poly_object.get_sollya_object()))
Log.report(
Log.Verbose, "sollya poly_object: {}".format(
poly_object.get_sollya_object()))
corr_exp = _vx_exp if exp_corr_factor == None else _vx_exp + exp_corr_factor
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
pre_result = -_log_inv_hi + (_red_vx * _poly + (-_log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = Conversion(corr_exp, precision=self.precision)
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_hex",
debug=debug_lftolx)
_result = exact_log2_hi_exp + pre_result
return _result, _poly, _log_inv_lo, _log_inv_hi, _red_vx
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
index_size = 3
approx_interval = Interval(0.0, 2**-index_size)
error_goal_approx = 2**-(self.precision.get_precision())
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
vx_int = Floor(vx * 2**index_size,
precision=self.precision,
tag="vx_int",
debug=debug_multi)
vx_frac = vx - (vx_int * 2**-index_size)
vx_frac.set_attributes(tag="vx_frac",
debug=debug_multi,
unbreakable=True)
poly_degree = sup(
guessdegree(2**(sollya.x), approx_interval, error_goal_approx)) + 1
precision_list = [1] + [self.precision] * (poly_degree)
vx_integer = Conversion(vx_int,
precision=int_precision,
tag="vx_integer",
debug=debug_multi)
vx_int_hi = BitLogicRightShift(vx_integer,
Constant(index_size),
tag="vx_int_hi",
debug=debug_multi)
vx_int_lo = Modulo(vx_integer,
2**index_size,
tag="vx_int_lo",
debug=debug_multi)
pow_exp = ExponentInsertion(Conversion(vx_int_hi,
precision=int_precision),
precision=self.precision,
tag="pow_exp",
debug=debug_multi)
exp2_table = ML_Table(dimensions=[2 * 2**index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("exp2_table"))
for i in range(2 * 2**index_size):
input_value = i - 2**index_size if i >= 2**index_size else i
exp2_value = SollyaObject(2)**((input_value) * 2**-index_size)
hi_value = round(exp2_value, self.precision.get_sollya_object(),
RN)
lo_value = round(exp2_value - hi_value,
self.precision.get_sollya_object(), RN)
exp2_table[i][0] = lo_value
exp2_table[i][1] = hi_value
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(
2**(sollya.x),
poly_degree,
precision_list,
approx_interval,
sollya.absolute,
error_function=error_function)
print "poly_approx_error: ", poly_approx_error, float(
log2(poly_approx_error))
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly = polynomial_scheme_builder(poly_object.sub_poly(start_index=1),
vx_frac,
unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
table_index = Addition(vx_int_lo,
Constant(2**index_size,
precision=int_precision),
precision=int_precision,
tag="table_index",
debug=debug_multi)
lo_value_load = TableLoad(exp2_table,
table_index,
0,
tag="lo_value_load",
debug=debug_multi)
hi_value_load = TableLoad(exp2_table,
table_index,
1,
tag="hi_value_load",
debug=debug_multi)
result = (hi_value_load +
(hi_value_load * poly +
(lo_value_load + lo_value_load * poly))) * pow_exp
ov_flag = Comparison(vx_int_hi,
Constant(self.precision.get_emax(),
precision=self.precision),
specifier=Comparison.Greater)
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = Statement(
Return(Select(ov_flag, FP_PlusInfty(self.precision), result)))
return scheme
def __init__(self,
precision=ML_Binary32,
abs_accuracy=S2**-24,
libm_compliant=True,
debug_flag=False,
fuse_fma=True,
fast_path_extract=True,
target=GenericProcessor(),
output_file="log1pf.c",
function_name="log1pf"):
# declaring CodeFunction and retrieving input variable
self.function_name = function_name
self.precision = precision
self.processor = target
func_implementation = CodeFunction(self.function_name,
output_format=self.precision)
vx = func_implementation.add_input_variable("x", self.precision)
sollya_precision = self.precision.sollya_object
# debug utilities
debugf = ML_Debug(display_format="%f")
debuglf = ML_Debug(display_format="%lf")
debugx = ML_Debug(display_format="%x")
debuglx = ML_Debug(display_format="%\"PRIx64\"", )
debugd = ML_Debug(display_format="%d",
pre_process=lambda v: "(int) %s" % v)
debugld = ML_Debug(display_format="%ld")
#debug_lftolx = ML_Debug(display_format = "%\"PRIx64\"", pre_process = lambda v: "double_to_64b_encoding(%s)" % v)
debug_lftolx = ML_Debug(
display_format="%\"PRIx64\" ev=%x",
pre_process=lambda v:
"double_to_64b_encoding(%s), __k1_fpu_get_exceptions()" % v)
debug_ddtolx = ML_Debug(
display_format="%\"PRIx64\" %\"PRIx64\"",
pre_process=lambda v:
"double_to_64b_encoding(%s.hi), double_to_64b_encoding(%s.lo)" %
(v, v))
debug_dd = ML_Debug(display_format="{.hi=%lf, .lo=%lf}",
pre_process=lambda v: "%s.hi, %s.lo" % (v, v))
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
log2_hi_value = round(
log(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(
log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = ML_Int64 if self.precision is ML_Binary64 else ML_Int32
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
log_table = ML_Table(dimensions=[2**table_index_size, 2],
storage_precision=self.precision)
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in xrange(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
inv_value = (1.0 + (inv_approx_table[i][0] / S2**9)) * S2**-1
value_high = round(
log(inv_value),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(
log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
# case close to 0: ctz
ctz_exp_limit = -7
ctz_cond = vx_exp < ctz_exp_limit
ctz_interval = Interval(-S2**ctz_exp_limit, S2**ctz_exp_limit)
ctz_poly_degree = sup(
guessdegree(
log1p(sollya.x) / sollya.x, ctz_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
ctz_poly_object = Polynomial.build_from_approximation(
log1p(sollya.x) / sollya.x, ctz_poly_degree,
[self.precision] * (ctz_poly_degree + 1), ctz_interval,
sollya.absolute)
print "generating polynomial evaluation scheme"
ctz_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
ctz_poly_object, vx, unified_precision=self.precision)
ctz_poly.set_attributes(tag="ctz_poly", debug=debug_lftolx)
ctz_result = vx * ctz_poly
neg_input = Comparison(vx,
-1,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
log_function_code = CodeFunction(
"new_log", [Variable("x", precision=ML_Binary64)],
output_format=ML_Binary64)
log_call_generator = FunctionOperator(
log_function_code.get_name(),
arity=1,
output_precision=ML_Binary64,
declare_prototype=log_function_code)
newlog_function = FunctionObject(log_function_code.get_name(),
(ML_Binary64, ), ML_Binary64,
log_call_generator)
# case away from 0.0
pre_vxp1 = vx + 1.0
pre_vxp1.set_attributes(tag="pre_vxp1", debug=debug_lftolx)
pre_vxp1_exp = ExponentExtraction(pre_vxp1,
tag="pre_vxp1_exp",
debug=debugd)
cm500 = Constant(-500, precision=ML_Int32)
c0 = Constant(0, precision=ML_Int32)
cond_scaling = pre_vxp1_exp > 2**(self.precision.get_exponent_size() -
2)
scaling_factor_exp = Select(cond_scaling, cm500, c0)
scaling_factor = ExponentInsertion(scaling_factor_exp,
precision=self.precision,
tag="scaling_factor")
vxp1 = pre_vxp1 * scaling_factor
vxp1.set_attributes(tag="vxp1", debug=debug_lftolx)
vxp1_exp = ExponentExtraction(vxp1, tag="vxp1_exp", debug=debugd)
vxp1_inv = DivisionSeed(vxp1,
precision=self.precision,
tag="vxp1_inv",
debug=debug_lftolx,
silent=True)
vxp1_dirty_inv = ExponentInsertion(-vxp1_exp,
precision=self.precision,
tag="vxp1_dirty_inv",
debug=debug_lftolx)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(vxp1, precision=int_precision, debug=debuglx),
self.precision.get_field_size() - 7,
debug=debuglx),
0x7f,
tag="table_index",
debug=debuglx)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(TypeCast(vxp1_inv,
precision=ML_UInt64),
Constant(-2,
precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
vxp1_dirty_inv,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
red_vxp1 = Select(cond_scaling, arg_red_index * vxp1 - 1.0,
(arg_red_index * vx - 1.0) + arg_red_index)
#red_vxp1 = arg_red_index * vxp1 - 1.0
red_vxp1.set_attributes(tag="red_vxp1", debug=debug_lftolx)
log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
print "building mathematical polynomial"
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
print "generating polynomial evaluation scheme"
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vxp1, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_lftolx)
print global_poly_object.get_sollya_object()
vxp1_inv_exp = ExponentExtraction(vxp1_inv,
tag="vxp1_inv_exp",
debug=debugd)
corr_exp = -vxp1_exp + scaling_factor_exp # vxp1_inv_exp
#poly = (red_vxp1) * (1 + _poly)
#poly.set_attributes(tag = "poly", debug = debug_lftolx, prevent_optimization = True)
pre_result = -log_inv_hi + (red_vxp1 + red_vxp1 * _poly +
(-corr_exp * log2_lo - log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = -corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_exp",
debug=debug_lftolx,
prevent_optimization=True)
#std_result = exact_log2_hi_exp + pre_result
exact_log2_lo_exp = -corr_exp * log2_lo
exact_log2_lo_exp.set_attributes(
tag="exact_log2_lo_exp",
debug=debug_lftolx) #, prevent_optimization = True)
init = exact_log2_lo_exp - log_inv_lo
init.set_attributes(tag="init",
debug=debug_lftolx,
prevent_optimization=True)
fma0 = (red_vxp1 * _poly + init) # - log_inv_lo)
fma0.set_attributes(tag="fma0", debug=debug_lftolx)
step0 = fma0
step0.set_attributes(
tag="step0", debug=debug_lftolx) #, prevent_optimization = True)
step1 = step0 + red_vxp1
step1.set_attributes(tag="step1",
debug=debug_lftolx,
prevent_optimization=True)
step2 = -log_inv_hi + step1
step2.set_attributes(tag="step2",
debug=debug_lftolx,
prevent_optimization=True)
std_result = exact_log2_hi_exp + step2
std_result.set_attributes(tag="std_result",
debug=debug_lftolx,
prevent_optimization=True)
# main scheme
print "MDL scheme"
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal, Return(vx),
ConditionBlock(ctz_cond, Statement(Return(ctz_result), ),
Statement(Return(std_result))))))
scheme = pre_scheme
#print scheme.get_str(depth = None, display_precision = True)
opt_eng = OptimizationEngine(self.processor)
# fusing FMA
print "MDL fusing FMA"
scheme = opt_eng.fuse_multiply_add(scheme, silence=True)
print "MDL abstract scheme"
opt_eng.instantiate_abstract_precision(scheme, None)
#print scheme.get_str(depth = None, display_precision = True)
print "MDL instantiated scheme"
opt_eng.instantiate_precision(scheme, default_precision=ML_Binary32)
print "subexpression sharing"
opt_eng.subexpression_sharing(scheme)
print "silencing operation"
opt_eng.silence_fp_operations(scheme)
# registering scheme as function implementation
func_implementation.set_scheme(scheme)
# check processor support
opt_eng.check_processor_support(scheme)
# factorizing fast path
opt_eng.factorize_fast_path(scheme)
#print scheme.get_str(depth = None, display_precision = True)
cg = CCodeGenerator(self.processor,
declare_cst=False,
disable_debug=not debug_flag,
libm_compliant=libm_compliant)
self.result = func_implementation.get_definition(cg,
C_Code,
static_cst=True)
self.result.add_header("support_lib/ml_special_values.h")
self.result.add_header("math.h")
self.result.add_header("stdio.h")
self.result.add_header("inttypes.h")
#print self.result.get(cg)
output_stream = open("%s.c" % func_implementation.get_name(), "w")
output_stream.write(self.result.get(cg))
output_stream.close()
def generate_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = Abs(self.implementation.add_input_variable("x", self.precision), tag = "vx")
Log.report(Log.Info, "generating implementation scheme")
if self.debug_flag:
Log.report(Log.Info, "debug has been enabled")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
debug_precision = {ML_Binary32: debug_ftox, ML_Binary64: debug_lftolx}[self.precision]
test_nan_or_inf = Test(vx, specifier = Test.IsInfOrNaN, likely = False, debug = True, tag = "nan_or_inf")
test_nan = Test(vx, specifier = Test.IsNaN, debug = True, tag = "is_nan_test")
test_positive = Comparison(vx, 0, specifier = Comparison.GreaterOrEqual, debug = True, tag = "inf_sign")
test_signaling_nan = Test(vx, specifier = Test.IsSignalingNaN, debug = True, tag = "is_signaling_nan")
return_snan = Statement(ExpRaiseReturn(ML_FPE_Invalid, return_value = FP_QNaN(self.precision)))
# return in case of infinity input
infty_return = Statement(ConditionBlock(test_positive, Return(FP_PlusInfty(self.precision)), Return(FP_PlusZero(self.precision))))
# return in case of specific value input (NaN or inf)
specific_return = ConditionBlock(test_nan, ConditionBlock(test_signaling_nan, return_snan, Return(FP_QNaN(self.precision))), infty_return)
# return in case of standard (non-special) input
sollya_precision = self.precision.get_sollya_object()
hi_precision = self.precision.get_field_size() - 3
# argument reduction
frac_pi_index = 3
frac_pi = round(S2**frac_pi_index / pi, sollya_precision, sollya.RN)
inv_frac_pi = round(pi / S2**frac_pi_index, hi_precision, sollya.RN)
inv_frac_pi_lo = round(pi / S2**frac_pi_index - inv_frac_pi, sollya_precision, sollya.RN)
# computing k = E(x * frac_pi)
vx_pi = Multiplication(vx, frac_pi, precision = self.precision)
k = NearestInteger(vx_pi, precision = ML_Int32, tag = "k", debug = True)
fk = Conversion(k, precision = self.precision, tag = "fk")
inv_frac_pi_cst = Constant(inv_frac_pi, tag = "inv_frac_pi", precision = self.precision)
inv_frac_pi_lo_cst = Constant(inv_frac_pi_lo, tag = "inv_frac_pi_lo", precision = self.precision)
red_vx_hi = (vx - inv_frac_pi_cst * fk)
red_vx_hi.set_attributes(tag = "red_vx_hi", debug = debug_precision, precision = self.precision)
red_vx_lo_sub = inv_frac_pi_lo_cst * fk
red_vx_lo_sub.set_attributes(tag = "red_vx_lo_sub", debug = debug_precision, unbreakable = True, precision = self.precision)
vx_d = Conversion(vx, precision = ML_Binary64, tag = "vx_d")
pre_red_vx = red_vx_hi - inv_frac_pi_lo_cst * fk
pre_red_vx_d_hi = (vx_d - inv_frac_pi_cst * fk)
pre_red_vx_d_hi.set_attributes(tag = "pre_red_vx_d_hi", precision = ML_Binary64, debug = debug_lftolx)
pre_red_vx_d = pre_red_vx_d_hi - inv_frac_pi_lo_cst * fk
pre_red_vx_d.set_attributes(tag = "pre_red_vx_d", debug = debug_lftolx, precision = ML_Binary64)
modk = Modulo(k, 2**(frac_pi_index+1), precision = ML_Int32, tag = "switch_value", debug = True)
sel_c = Equal(BitLogicAnd(modk, 2**(frac_pi_index-1)), 2**(frac_pi_index-1))
red_vx = Select(sel_c, -pre_red_vx, pre_red_vx)
red_vx.set_attributes(tag = "red_vx", debug = debug_precision, precision = self.precision)
red_vx_d = Select(sel_c, -pre_red_vx_d, pre_red_vx_d)
red_vx_d.set_attributes(tag = "red_vx_d", debug = debug_lftolx, precision = ML_Binary64)
approx_interval = Interval(-pi/(S2**(frac_pi_index+1)), pi / S2**(frac_pi_index+1))
Log.report(Log.Info, "approx interval: %s\n" % approx_interval)
error_goal_approx = S2**-self.precision.get_precision()
Log.report(Log.Info, "building mathematical polynomial")
poly_degree_vector = [None] * 2**(frac_pi_index+1)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
#polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
index_relative = []
poly_object_vector = [None] * 2**(frac_pi_index+1)
for i in range(2**(frac_pi_index+1)):
sub_func = cos(sollya.x+i*pi/S2**frac_pi_index)
degree = int(sup(guessdegree(sub_func, approx_interval, error_goal_approx))) + 1
degree_list = range(degree+1)
a_interval = approx_interval
if i == 0:
# ad-hoc, TODO: to be cleaned
degree = 6
degree_list = range(0, degree+1, 2)
elif i % 2**(frac_pi_index) == 2**(frac_pi_index-1):
# for pi/2 and 3pi/2, an approx to sin=cos(pi/2+x)
# must be generated
degree_list = range(1, degree+1, 2)
if i == 3 or i == 5 or i == 7 or i == 9:
precision_list = [sollya.binary64] + [sollya.binary32] *(degree)
else:
precision_list = [sollya.binary32] * (degree+1)
poly_degree_vector[i] = degree
constraint = sollya.absolute
delta = (2**(frac_pi_index - 3))
centered_i = (i % 2**(frac_pi_index)) - 2**(frac_pi_index-1)
if centered_i < delta and centered_i > -delta and centered_i != 0:
constraint = sollya.relative
index_relative.append(i)
Log.report(Log.Info, "generating approximation for %d/%d" % (i, 2**(frac_pi_index+1)))
poly_object_vector[i], _ = Polynomial.build_from_approximation_with_error(sub_func, degree_list, precision_list, a_interval, constraint, error_function = error_function)
# unified power map for red_sx^n
upm = {}
rel_error_list = []
poly_scheme_vector = [None] * (2**(frac_pi_index+1))
for i in range(2**(frac_pi_index+1)):
poly_object = poly_object_vector[i]
poly_precision = self.precision
if i == 3 or i == 5 or i == 7 or i == 9:
poly_precision = ML_Binary64
c0 = Constant(coeff(poly_object.get_sollya_object(), 0), precision = ML_Binary64)
c1 = Constant(coeff(poly_object.get_sollya_object(), 1), precision = self.precision)
poly_hi = (c0 + c1 * red_vx)
poly_hi.set_precision(ML_Binary64)
red_vx_d_2 = red_vx_d * red_vx_d
poly_scheme = poly_hi + red_vx_d_2 * polynomial_scheme_builder(poly_object.sub_poly(start_index = 2, offset = 2), red_vx, unified_precision = self.precision, power_map_ = upm)
poly_scheme.set_attributes(unbreakable = True)
elif i == 4:
c1 = Constant(coeff(poly_object.get_sollya_object(), 1), precision = ML_Binary64)
poly_scheme = c1 * red_vx_d + polynomial_scheme_builder(poly_object.sub_poly(start_index = 2), red_vx, unified_precision = self.precision, power_map_ = upm)
poly_scheme.set_precision(ML_Binary64)
else:
poly_scheme = polynomial_scheme_builder(poly_object, red_vx, unified_precision = poly_precision, power_map_ = upm)
#if i == 3:
# c0 = Constant(coeff(poly_object.get_sollya_object(), 0), precision = self.precision)
# c1 = Constant(coeff(poly_object.get_sollya_object(), 1), precision = self.precision)
# poly_scheme = (c0 + c1 * red_vx) + polynomial_scheme_builder(poly_object.sub_poly(start_index = 2), red_vx, unified_precision = self.precision, power_map_ = upm)
poly_scheme.set_attributes(tag = "poly_cos%dpi%d" % (i, 2**(frac_pi_index)), debug = debug_precision)
poly_scheme_vector[i] = poly_scheme
#try:
if is_gappa_installed() and i == 3:
opt_scheme = self.opt_engine.optimization_process(poly_scheme, self.precision, copy = True, fuse_fma = self.fuse_fma)
tag_map = {}
self.opt_engine.register_nodes_by_tag(opt_scheme, tag_map)
gappa_vx = Variable("red_vx", precision = self.precision, interval = approx_interval)
cg_eval_error_copy_map = {
tag_map["red_vx"]: gappa_vx,
tag_map["red_vx_d"]: gappa_vx,
}
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
if self.libm_compliant:
return RaiseReturn(*args, precision=self.precision, **kwords)
else:
return Return(kwords["return_value"], precision=self.precision)
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=debug_multi,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=debug_multi,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=debug_multi,
tag="is_signaling_nan")
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
# return in case of infinity input
infty_return = Statement(
ConditionBlock(
test_positive,
Return(FP_PlusInfty(self.precision), precision=self.precision),
Return(FP_PlusZero(self.precision), precision=self.precision)))
# return in case of specific value input (NaN or inf)
specific_return = ConditionBlock(
test_nan,
ConditionBlock(
test_signaling_nan, return_snan,
Return(FP_QNaN(self.precision), precision=self.precision)),
infty_return)
# return in case of standard (non-special) input
# exclusion of early overflow and underflow cases
precision_emax = self.precision.get_emax()
precision_max_value = S2 * S2**precision_emax
exp_overflow_bound = sollya.ceil(log(precision_max_value))
early_overflow_test = Comparison(vx,
exp_overflow_bound,
likely=False,
specifier=Comparison.Greater)
early_overflow_return = Statement(
ClearException() if self.libm_compliant else Statement(),
ExpRaiseReturn(ML_FPE_Inexact,
ML_FPE_Overflow,
return_value=FP_PlusInfty(self.precision)))
precision_emin = self.precision.get_emin_subnormal()
precision_min_value = S2**precision_emin
exp_underflow_bound = floor(log(precision_min_value))
early_underflow_test = Comparison(vx,
exp_underflow_bound,
likely=False,
specifier=Comparison.Less)
early_underflow_return = Statement(
ClearException() if self.libm_compliant else Statement(),
ExpRaiseReturn(ML_FPE_Inexact,
ML_FPE_Underflow,
return_value=FP_PlusZero(self.precision)))
# constant computation
invlog2 = self.precision.round_sollya_object(1 / log(2), sollya.RN)
interval_vx = Interval(exp_underflow_bound, exp_overflow_bound)
interval_fk = interval_vx * invlog2
interval_k = Interval(floor(inf(interval_fk)),
sollya.ceil(sup(interval_fk)))
log2_hi_precision = self.precision.get_field_size() - (
sollya.ceil(log2(sup(abs(interval_k)))) + 2)
Log.report(Log.Info, "log2_hi_precision: %d" % log2_hi_precision)
invlog2_cst = Constant(invlog2, precision=self.precision)
log2_hi = round(log(2), log2_hi_precision, sollya.RN)
log2_lo = self.precision.round_sollya_object(
log(2) - log2_hi, sollya.RN)
# argument reduction
unround_k = vx * invlog2
unround_k.set_attributes(tag="unround_k", debug=debug_multi)
k = NearestInteger(unround_k,
precision=self.precision,
debug=debug_multi)
ik = NearestInteger(unround_k,
precision=self.precision.get_integer_format(),
debug=debug_multi,
tag="ik")
ik.set_tag("ik")
k.set_tag("k")
exact_pre_mul = (k * log2_hi)
exact_pre_mul.set_attributes(exact=True)
exact_hi_part = vx - exact_pre_mul
exact_hi_part.set_attributes(exact=True,
tag="exact_hi",
debug=debug_multi,
prevent_optimization=True)
exact_lo_part = -k * log2_lo
exact_lo_part.set_attributes(tag="exact_lo",
debug=debug_multi,
prevent_optimization=True)
r = exact_hi_part + exact_lo_part
r.set_tag("r")
r.set_attributes(debug=debug_multi)
approx_interval = Interval(-log(2) / 2, log(2) / 2)
approx_interval_half = approx_interval / 2
approx_interval_split = [
Interval(-log(2) / 2, inf(approx_interval_half)),
approx_interval_half,
Interval(sup(approx_interval_half),
log(2) / 2)
]
# TODO: should be computed automatically
exact_hi_interval = approx_interval
exact_lo_interval = -interval_k * log2_lo
opt_r = self.optimise_scheme(r, copy={})
tag_map = {}
self.opt_engine.register_nodes_by_tag(opt_r, tag_map)
cg_eval_error_copy_map = {
vx:
Variable("x", precision=self.precision, interval=interval_vx),
tag_map["k"]:
Variable("k", interval=interval_k, precision=self.precision)
}
#try:
if is_gappa_installed():
eval_error = self.gappa_engine.get_eval_error_v2(
self.opt_engine,
opt_r,
cg_eval_error_copy_map,
gappa_filename="red_arg.g")
else:
eval_error = 0.0
Log.report(Log.Warning,
"gappa is not installed in this environnement")
Log.report(Log.Info, "eval error: %s" % eval_error)
local_ulp = sup(ulp(sollya.exp(approx_interval), self.precision))
# FIXME refactor error_goal from accuracy
Log.report(Log.Info, "accuracy: %s" % self.accuracy)
if isinstance(self.accuracy, ML_Faithful):
error_goal = local_ulp
elif isinstance(self.accuracy, ML_CorrectlyRounded):
error_goal = S2**-1 * local_ulp
elif isinstance(self.accuracy, ML_DegradedAccuracyAbsolute):
error_goal = self.accuracy.goal
elif isinstance(self.accuracy, ML_DegradedAccuracyRelative):
error_goal = self.accuracy.goal
else:
Log.report(Log.Error, "unknown accuracy: %s" % self.accuracy)
# error_goal = local_ulp #S2**-(self.precision.get_field_size()+1)
error_goal_approx = S2**-1 * error_goal
Log.report(Log.Info,
"\033[33;1m building mathematical polynomial \033[0m\n")
poly_degree = max(
sup(
guessdegree(
expm1(sollya.x) / sollya.x, approx_interval,
error_goal_approx)) - 1, 2)
init_poly_degree = poly_degree
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme
#polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
while 1:
Log.report(Log.Info, "attempting poly degree: %d" % poly_degree)
precision_list = [1] + [self.precision] * (poly_degree)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(
expm1(sollya.x),
poly_degree,
precision_list,
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(Log.Info, "polynomial: %s " % poly_object)
sub_poly = poly_object.sub_poly(start_index=2)
Log.report(Log.Info, "polynomial: %s " % sub_poly)
Log.report(Log.Info, "poly approx error: %s" % poly_approx_error)
Log.report(
Log.Info,
"\033[33;1m generating polynomial evaluation scheme \033[0m")
pre_poly = polynomial_scheme_builder(
poly_object, r, unified_precision=self.precision)
pre_poly.set_attributes(tag="pre_poly", debug=debug_multi)
pre_sub_poly = polynomial_scheme_builder(
sub_poly, r, unified_precision=self.precision)
pre_sub_poly.set_attributes(tag="pre_sub_poly", debug=debug_multi)
poly = 1 + (exact_hi_part + (exact_lo_part + pre_sub_poly))
poly.set_tag("poly")
# optimizing poly before evaluation error computation
#opt_poly = self.opt_engine.optimization_process(poly, self.precision, fuse_fma = fuse_fma)
#opt_sub_poly = self.opt_engine.optimization_process(pre_sub_poly, self.precision, fuse_fma = fuse_fma)
opt_poly = self.optimise_scheme(poly)
opt_sub_poly = self.optimise_scheme(pre_sub_poly)
# evaluating error of the polynomial approximation
r_gappa_var = Variable("r",
precision=self.precision,
interval=approx_interval)
exact_hi_gappa_var = Variable("exact_hi",
precision=self.precision,
interval=exact_hi_interval)
exact_lo_gappa_var = Variable("exact_lo",
precision=self.precision,
interval=exact_lo_interval)
vx_gappa_var = Variable("x",
precision=self.precision,
interval=interval_vx)
k_gappa_var = Variable("k",
interval=interval_k,
precision=self.precision)
#print "exact_hi interval: ", exact_hi_interval
sub_poly_error_copy_map = {
#r.get_handle().get_node(): r_gappa_var,
#vx.get_handle().get_node(): vx_gappa_var,
exact_hi_part.get_handle().get_node():
exact_hi_gappa_var,
exact_lo_part.get_handle().get_node():
exact_lo_gappa_var,
#k.get_handle().get_node(): k_gappa_var,
}
poly_error_copy_map = {
exact_hi_part.get_handle().get_node(): exact_hi_gappa_var,
exact_lo_part.get_handle().get_node(): exact_lo_gappa_var,
}
if is_gappa_installed():
sub_poly_eval_error = -1.0
sub_poly_eval_error = self.gappa_engine.get_eval_error_v2(
self.opt_engine,
opt_sub_poly,
sub_poly_error_copy_map,
gappa_filename="%s_gappa_sub_poly.g" % self.function_name)
dichotomy_map = [
{
exact_hi_part.get_handle().get_node():
approx_interval_split[0],
},
{
exact_hi_part.get_handle().get_node():
approx_interval_split[1],
},
{
exact_hi_part.get_handle().get_node():
approx_interval_split[2],
},
]
poly_eval_error_dico = self.gappa_engine.get_eval_error_v3(
self.opt_engine,
opt_poly,
poly_error_copy_map,
gappa_filename="gappa_poly.g",
dichotomy=dichotomy_map)
poly_eval_error = max(
[sup(abs(err)) for err in poly_eval_error_dico])
else:
poly_eval_error = 0.0
sub_poly_eval_error = 0.0
Log.report(Log.Warning,
"gappa is not installed in this environnement")
Log.report(Log.Info, "stopping autonomous degree research")
# incrementing polynomial degree to counteract initial decrementation effect
poly_degree += 1
break
Log.report(Log.Info, "poly evaluation error: %s" % poly_eval_error)
Log.report(Log.Info,
"sub poly evaluation error: %s" % sub_poly_eval_error)
global_poly_error = None
global_rel_poly_error = None
for case_index in range(3):
poly_error = poly_approx_error + poly_eval_error_dico[
case_index]
rel_poly_error = sup(
abs(poly_error /
sollya.exp(approx_interval_split[case_index])))
if global_rel_poly_error == None or rel_poly_error > global_rel_poly_error:
global_rel_poly_error = rel_poly_error
global_poly_error = poly_error
flag = error_goal > global_rel_poly_error
if flag:
break
else:
poly_degree += 1
late_overflow_test = Comparison(ik,
self.precision.get_emax(),
specifier=Comparison.Greater,
likely=False,
debug=debug_multi,
tag="late_overflow_test")
overflow_exp_offset = (self.precision.get_emax() -
self.precision.get_field_size() / 2)
diff_k = Subtraction(
ik,
Constant(overflow_exp_offset,
precision=self.precision.get_integer_format()),
precision=self.precision.get_integer_format(),
debug=debug_multi,
tag="diff_k",
)
late_overflow_result = (ExponentInsertion(
diff_k, precision=self.precision) * poly) * ExponentInsertion(
overflow_exp_offset, precision=self.precision)
late_overflow_result.set_attributes(silent=False,
tag="late_overflow_result",
debug=debug_multi,
precision=self.precision)
late_overflow_return = ConditionBlock(
Test(late_overflow_result, specifier=Test.IsInfty, likely=False),
ExpRaiseReturn(ML_FPE_Overflow,
return_value=FP_PlusInfty(self.precision)),
Return(late_overflow_result, precision=self.precision))
late_underflow_test = Comparison(k,
self.precision.get_emin_normal(),
specifier=Comparison.LessOrEqual,
likely=False)
underflow_exp_offset = 2 * self.precision.get_field_size()
corrected_exp = Addition(
ik,
Constant(underflow_exp_offset,
precision=self.precision.get_integer_format()),
precision=self.precision.get_integer_format(),
tag="corrected_exp")
late_underflow_result = (
ExponentInsertion(corrected_exp, precision=self.precision) *
poly) * ExponentInsertion(-underflow_exp_offset,
precision=self.precision)
late_underflow_result.set_attributes(debug=debug_multi,
tag="late_underflow_result",
silent=False)
test_subnormal = Test(late_underflow_result,
specifier=Test.IsSubnormal)
late_underflow_return = Statement(
ConditionBlock(
test_subnormal,
ExpRaiseReturn(ML_FPE_Underflow,
return_value=late_underflow_result)),
Return(late_underflow_result, precision=self.precision))
twok = ExponentInsertion(ik,
tag="exp_ik",
debug=debug_multi,
precision=self.precision)
#std_result = twok * ((1 + exact_hi_part * pre_poly) + exact_lo_part * pre_poly)
std_result = twok * poly
std_result.set_attributes(tag="std_result", debug=debug_multi)
result_scheme = ConditionBlock(
late_overflow_test, late_overflow_return,
ConditionBlock(late_underflow_test, late_underflow_return,
Return(std_result, precision=self.precision)))
std_return = ConditionBlock(
early_overflow_test, early_overflow_return,
ConditionBlock(early_underflow_test, early_underflow_return,
result_scheme))
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = ConditionBlock(
test_nan_or_inf,
Statement(ClearException() if self.libm_compliant else Statement(),
specific_return), std_return)
return scheme
Log.report(Log.Error, "poly degree must be positive or null. {}, {}",
poly_object.degree, poly_object)
if __name__ == "__main__":
implem_results = []
for eps_target in [S2**-40, S2**-50, S2**-55, S2**-60, S2**-65]:
approx_interval = Interval(-S2**-5, S2**-5)
ctx = MLL_Context(ML_Binary64, approx_interval)
vx = Variable("x",
precision=ctx.variableFormat,
interval=approx_interval)
# guessding the best degree
poly_degree = int(
sup(
sollya.guessdegree(sollya.exp(sollya.x), approx_interval,
eps_target)))
# asking sollya to provide the approximation
poly_object = Polynomial.build_from_approximation(
sollya.exp(sollya.x), poly_degree,
[sollya.doubledouble] * (poly_degree + 1), vx.interval)
print("poly object is {}".format(poly_object))
poly_graph, poly_epsilon = mll_implementpoly_horner(
ctx, poly_object, eps_target, vx)
print("poly_graph is {}".format(
poly_graph.get_str(depth=None, display_precision=True)))
print("poly epsilon is {}".format(float(poly_epsilon)))
print("poly accuracy is {}".format(
get_accuracy_from_epsilon(poly_epsilon)))
implem_results.append(
(eps_target, poly_degree, poly_object, poly_graph, poly_epsilon))