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 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
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)
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 piecewise_approximation(function,
variable,
precision,
bound_low=-1.0,
bound_high=1.0,
num_intervals=16,
max_degree=2,
error_threshold=S2**-24,
odd=False,
even=False):
""" Generate a piecewise approximation
:param function: function to be approximated
:type function: SollyaObject
:param variable: input variable
:type variable: Variable
:param precision: variable's format
:type precision: ML_Format
:param bound_low: lower bound for the approximation interval
:param bound_high: upper bound for the approximation interval
:param num_intervals: number of sub-interval / sub-division of the main interval
:param max_degree: maximum degree for an approximation on any sub-interval
:param error_threshold: error bound for an approximation on any sub-interval
:return: pair (scheme, error) where scheme is a graph node for an
approximation scheme of function evaluated at variable, and error
is the maximum approximation error encountered
:rtype tuple(ML_Operation, SollyaObject): """
degree_generator = piecewise_approximation_degree_generator(
function,
bound_low,
bound_high,
num_intervals=num_intervals,
error_threshold=error_threshold,
)
degree_list = list(degree_generator)
# if max_degree is None then we determine it locally
if max_degree is None:
max_degree = max(degree_list)
# table to store coefficients of the approximation on each segment
coeff_table = ML_NewTable(
dimensions=[num_intervals, max_degree + 1],
storage_precision=precision,
tag="coeff_table",
const=True # by default all approximation coeff table are const
)
error_function = lambda p, f, ai, mod, t: sollya.dirtyinfnorm(p - f, ai)
max_approx_error = 0.0
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 = degree_list[i]
if local_degree > max_degree:
Log.report(
Log.Warning,
"local_degree {} exceeds max_degree bound ({}) in piecewise_approximation",
local_degree, max_degree)
# as max_degree defines the size of the table we can use
# it as the degree for each sub-interval polynomial
# as there is nothing to gain (yet) by using a smaller polynomial
degree = max_degree # min(max_degree, local_degree)
if function(subint_low) == 0.0:
# if the lower bound is a zero to the function, we
# need to force value=0 for the constant coefficient
# and extend the approximation interval
local_poly_degree_list = list(
range(1 if even else 0, degree + 1, 2 if odd or even else 1))
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
function(sollya.x) / sollya.x,
local_poly_degree_list,
[precision] * len(local_poly_degree_list),
Interval(-subint_high * 0.95, subint_high),
sollya.absolute,
error_function=error_function)
# multiply by sollya.x
poly_object = poly_object.sub_poly(offset=-1)
else:
try:
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
local_function,
degree, [precision] * (degree + 1),
local_interval,
sollya.absolute,
error_function=error_function)
except SollyaError as err:
# try to see if function is constant on the interval (possible
# failure cause for fpminmax)
cst_value = precision.round_sollya_object(
function(subint_low), sollya.RN)
accuracy = error_threshold
diff_with_cst_range = sollya.supnorm(cst_value, local_function,
local_interval,
sollya.absolute, accuracy)
diff_with_cst = sup(abs(diff_with_cst_range))
if diff_with_cst < error_threshold:
Log.report(Log.Info, "constant polynomial detected")
poly_object = Polynomial([function(subint_low)] +
[0] * degree)
approx_error = diff_with_cst
else:
Log.report(
Log.error,
"degree: {} for index {}, diff_with_cst={} (vs error_threshold={}) ",
degree,
i,
diff_with_cst,
error_threshold,
error=err)
for ci in range(max_degree + 1):
if ci in poly_object.coeff_map:
coeff_table[i][ci] = poly_object.coeff_map[ci]
else:
coeff_table[i][ci] = 0.0
if approx_error > error_threshold:
Log.report(
Log.Warning,
"piecewise_approximation on index {} exceeds error threshold: {} > {}",
i, approx_error, error_threshold)
max_approx_error = max(max_approx_error, abs(approx_error))
# computing offset
diff = Subtraction(variable,
Constant(bound_low, precision=precision),
tag="diff",
debug=debug_multi,
precision=precision)
int_prec = precision.get_integer_format()
# delta = bound_high - bound_low
delta_ratio = Constant(num_intervals / (bound_high - bound_low),
precision=precision)
# computing table index
# index = nearestint(diff / delta * <num_intervals>)
index = Max(0,
Min(
NearestInteger(
Multiplication(diff, delta_ratio, precision=precision),
precision=int_prec,
), num_intervals - 1),
tag="index",
debug=debug_multi,
precision=int_prec)
poly_var = Subtraction(diff,
Multiplication(
Conversion(index, precision=precision),
Constant(interval_size, precision=precision)),
precision=precision,
tag="poly_var",
debug=debug_multi)
# generating indexed polynomial
coeffs = [(ci, TableLoad(coeff_table, index, ci))
for ci in range(max_degree + 1)][::-1]
poly_scheme = PolynomialSchemeEvaluator.generate_horner_scheme2(
coeffs, poly_var, precision, {}, precision)
return poly_scheme, max_approx_error
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_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 = 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 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_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = self.implementation.add_input_variable("x", self.precision)
table_size_log = self.table_size_log
integer_size = 31
integer_precision = ML_Int32
max_bound = sup(abs(self.input_intervals[0]))
max_bound_log = int(ceil(log2(max_bound)))
Log.report(Log.Info, "max_bound_log=%s " % max_bound_log)
scaling_power = integer_size - max_bound_log
Log.report(Log.Info, "scaling power: %s " % scaling_power)
storage_precision = ML_Custom_FixedPoint_Format(1, 30, signed=True)
Log.report(Log.Info, "tabulating cosine and sine")
# cosine and sine fused table
fused_table = ML_NewTable(
dimensions=[2**table_size_log, 2],
storage_precision=storage_precision,
tag="fast_lib_shared_table") # self.uniquify_name("cossin_table"))
# filling table
for i in range(2**table_size_log):
local_x = i / S2**table_size_log * S2**max_bound_log
cos_local = cos(
local_x
) # nearestint(cos(local_x) * S2**storage_precision.get_frac_size())
sin_local = sin(
local_x
) # nearestint(sin(local_x) * S2**storage_precision.get_frac_size())
fused_table[i][0] = cos_local
fused_table[i][1] = sin_local
# argument reduction evaluation scheme
# scaling_factor = Constant(S2**scaling_power, precision = self.precision)
red_vx_precision = ML_Custom_FixedPoint_Format(31 - scaling_power,
scaling_power,
signed=True)
Log.report(
Log.Verbose, "red_vx_precision.get_c_bit_size()=%d" %
red_vx_precision.get_c_bit_size())
# red_vx = NearestInteger(vx * scaling_factor, precision = integer_precision)
red_vx = Conversion(vx,
precision=red_vx_precision,
tag="red_vx",
debug=debug_fixed32)
computation_precision = red_vx_precision # self.precision
output_precision = self.get_output_precision()
Log.report(Log.Info,
"computation_precision is %s" % computation_precision)
Log.report(Log.Info, "storage_precision is %s" % storage_precision)
Log.report(Log.Info, "output_precision is %s" % output_precision)
hi_mask_value = 2**32 - 2**(32 - table_size_log - 1)
hi_mask = Constant(hi_mask_value, precision=ML_Int32)
Log.report(Log.Info, "hi_mask=0x%x" % hi_mask_value)
red_vx_hi_int = BitLogicAnd(TypeCast(red_vx, precision=ML_Int32),
hi_mask,
precision=ML_Int32,
tag="red_vx_hi_int",
debug=debugd)
red_vx_hi = TypeCast(red_vx_hi_int,
precision=red_vx_precision,
tag="red_vx_hi",
debug=debug_fixed32)
red_vx_lo = red_vx - red_vx_hi
red_vx_lo.set_attributes(precision=red_vx_precision,
tag="red_vx_lo",
debug=debug_fixed32)
table_index = BitLogicRightShift(TypeCast(red_vx, precision=ML_Int32),
scaling_power -
(table_size_log - max_bound_log),
precision=ML_Int32,
tag="table_index",
debug=debugd)
tabulated_cos = TableLoad(fused_table,
table_index,
0,
tag="tab_cos",
precision=storage_precision,
debug=debug_fixed32)
tabulated_sin = TableLoad(fused_table,
table_index,
1,
tag="tab_sin",
precision=storage_precision,
debug=debug_fixed32)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
Log.report(Log.Info, "building polynomial approximation for cosine")
# cosine polynomial approximation
poly_interval = Interval(0, S2**(max_bound_log - table_size_log))
Log.report(Log.Info, "poly_interval=%s " % poly_interval)
cos_poly_degree = 2 # int(sup(guessdegree(cos(x), poly_interval, accuracy_goal)))
Log.report(Log.Verbose, "cosine polynomial approximation")
cos_poly_object, cos_approx_error = Polynomial.build_from_approximation_with_error(
cos(sollya.x), [0, 2],
[0] + [computation_precision.get_bit_size()],
poly_interval,
sollya.absolute,
error_function=error_function)
#cos_eval_scheme = PolynomialSchemeEvaluator.generate_horner_scheme(cos_poly_object, red_vx_lo, unified_precision = computation_precision)
Log.report(Log.Info, "cos_approx_error=%e" % cos_approx_error)
cos_coeff_list = cos_poly_object.get_ordered_coeff_list()
coeff_C0 = cos_coeff_list[0][1]
coeff_C2 = Constant(cos_coeff_list[1][1],
precision=ML_Custom_FixedPoint_Format(-1,
32,
signed=True))
Log.report(Log.Info, "building polynomial approximation for sine")
# sine polynomial approximation
sin_poly_degree = 2 # int(sup(guessdegree(sin(x)/x, poly_interval, accuracy_goal)))
Log.report(Log.Info, "sine poly degree: %e" % sin_poly_degree)
Log.report(Log.Verbose, "sine polynomial approximation")
sin_poly_object, sin_approx_error = Polynomial.build_from_approximation_with_error(
sin(sollya.x) / sollya.x, [0, 2], [0] +
[computation_precision.get_bit_size()] * (sin_poly_degree + 1),
poly_interval,
sollya.absolute,
error_function=error_function)
sin_coeff_list = sin_poly_object.get_ordered_coeff_list()
coeff_S0 = sin_coeff_list[0][1]
coeff_S2 = Constant(sin_coeff_list[1][1],
precision=ML_Custom_FixedPoint_Format(-1,
32,
signed=True))
# scheme selection between sine and cosine
if self.cos_output:
scheme = self.generate_cos_scheme(computation_precision,
tabulated_cos, tabulated_sin,
coeff_S2, coeff_C2, red_vx_lo)
else:
scheme = self.generate_sin_scheme(computation_precision,
tabulated_cos, tabulated_sin,
coeff_S2, coeff_C2, red_vx_lo)
result = Conversion(scheme, precision=self.get_output_precision())
Log.report(
Log.Verbose, "result operation tree :\n %s " % result.get_str(
display_precision=True, depth=None, memoization_map={}))
scheme = Statement(Return(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,
}
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 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 piecewise_approximation(function,
variable,
precision,
bound_low=-1.0,
bound_high=1.0,
num_intervals=16,
max_degree=2,
error_threshold=sollya.S2**-24):
""" To be documented """
# table to store coefficients of the approximation on each segment
coeff_table = ML_NewTable(dimensions=[num_intervals, max_degree + 1],
storage_precision=precision,
tag="coeff_table")
error_function = lambda p, f, ai, mod, t: sollya.dirtyinfnorm(p - f, ai)
max_approx_error = 0.0
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)
#local_interval = Interval(subint_low, subint_high)
local_function = function(sollya.x + subint_low)
local_interval = Interval(-interval_size, interval_size)
local_degree = sollya.guessdegree(local_function, local_interval,
error_threshold)
degree = min(max_degree, local_degree)
if function(subint_low) == 0.0:
# if the lower bound is a zero to the function, we
# need to force value=0 for the constant coefficient
# and extend the approximation interval
degree_list = range(1, degree + 1)
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
function(sollya.x),
degree_list, [precision] * len(degree_list),
Interval(-subint_high, subint_high),
sollya.absolute,
error_function=error_function)
else:
try:
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
local_function,
degree, [precision] * (degree + 1),
local_interval,
sollya.absolute,
error_function=error_function)
except SollyaError as err:
print("degree: {}".format(degree))
raise err
for ci in range(degree + 1):
if ci in poly_object.coeff_map:
coeff_table[i][ci] = poly_object.coeff_map[ci]
else:
coeff_table[i][ci] = 0.0
max_approx_error = max(max_approx_error, abs(approx_error))
# computing offset
diff = Subtraction(variable,
Constant(bound_low, precision=precision),
tag="diff",
precision=precision)
# delta = bound_high - bound_low
delta_ratio = Constant(num_intervals / (bound_high - bound_low),
precision=precision)
# computing table index
# index = nearestint(diff / delta * <num_intervals>)
index = Max(0,
Min(
NearestInteger(Multiplication(diff,
delta_ratio,
precision=precision),
precision=ML_Int32), num_intervals - 1),
tag="index",
debug=True,
precision=ML_Int32)
poly_var = Subtraction(diff,
Multiplication(
Conversion(index, precision=precision),
Constant(interval_size, precision=precision)),
precision=precision,
tag="poly_var",
debug=True)
# generating indexed polynomial
coeffs = [(ci, TableLoad(coeff_table, index, ci))
for ci in range(degree + 1)][::-1]
poly_scheme = PolynomialSchemeEvaluator.generate_horner_scheme2(
coeffs, poly_var, precision, {}, precision)
return poly_scheme, max_approx_error
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
vx = Abs(vx)
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
# argument reduction
arg_reg_value = log(2) / 2**index_size
inv_log2_value = round(1 / arg_reg_value,
self.precision.get_sollya_object(), 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(2^21024)
# 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, RN)
log2_lo_value = round(arg_reg_value - log2_hi_value,
self.precision.get_sollya_object(), 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)
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=int_precision)
})
print "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))
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),
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_Table(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
# 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, self.precision.get_sollya_object(),
RN)
pos_value_lo = round(exp_value - pos_value_hi,
self.precision.get_sollya_object(), RN)
neg_value_hi = round(mexp_value,
self.precision.get_sollya_object(), RN)
neg_value_lo = round(mexp_value - neg_value_hi,
self.precision.get_sollya_object(), 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
# cosh(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
# exp(x) = exp(r) * 2^h * 2^(l *2^-index_size)
#
# cosh(x) = exp(r) * 2^(h-1) 2^(l *2^-index_size) + exp(-r) * 2^(-h-1) * 2^(-l *2^-index_size)
#
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)
print "poly_approx_error: ", 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))
pow_exp_pos = ExponentInsertion(k_plus, precision=self.precision)
pow_exp_neg = ExponentInsertion(k_neg, precision=self.precision)
pos_exp = (
pos_value_load_hi +
(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 +
(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(pos_exp,
neg_exp,
precision=self.precision,
tag="result",
debug=debug_multi)
# ov_value
ov_value = round(acosh(self.precision.get_max_value()),
self.precision.get_sollya_object(), RD)
ov_flag = Comparison(Abs(vx),
Constant(ov_value, 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
