def generate_fptaylor(x):
x_low = sollya.inf(x)
x_high = sollya.sup(x)
query = "\n".join([
"Variables", " real x in [{},{}];".format(x_low, x_high),
"Definitions", " r rnd64= x;",
" retval rnd64= {};".format(poly_expr), "Expressions",
" retval;"
])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "true",
"--abs-error": "true"
})
rnd_rel_err = float(
res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "false",
"--abs-error": "true"
})
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.exp(sollya.x), x, sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.exp(sollya.x), x, sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def get_default_args(**kw):
""" Return a structure containing the arguments for ML_GenericLog,
builtin from a default argument mapping overloaded with @p kw """
default_args_log = {
"output_file": "ml_genlog.c",
"function_name": "ml_genlog",
"precision": ML_Binary32,
"accuracy": ML_Faithful,
"target": GenericProcessor(),
"basis": exp(1),
}
default_args_log.update(kw)
return DefaultArgTemplate(**default_args_log)
def generate_scheme(self):
x = self.implementation.add_input_variable("x", self.precision)
n_log2 = self.precision.round_sollya_object(sollya.log(2), sollya.RN)
if not self.skip_reduction:
n_invlog2 = self.precision.round_sollya_object(
1 / sollya.log(2), sollya.RN)
invlog2 = Constant(n_invlog2, tag="invlog2")
unround_k = Multiplication(x, invlog2, tag="unround_k")
k = Floor(unround_k, precision=self.precision, tag="k")
log2 = Constant(n_log2, tag="log2")
whole = Multiplication(k, log2, tag="whole")
r = Subtraction(x, whole, tag="r")
else:
r = x
approx_interval = sollya.Interval(-2**-10, n_log2 + 2**-10)
approx_func = sollya.exp(sollya.x)
builder = Polynomial.build_from_approximation
sollya.settings.prec = 2**10
poly_object = builder(approx_func, self.poly_degree,
[self.precision] * (self.poly_degree + 1),
approx_interval, sollya.relative)
self.poly_object = poly_object
schemer = PolynomialSchemeEvaluator.generate_horner_scheme
poly = schemer(poly_object, r)
if not self.skip_reduction:
ik = Conversion(k,
precision=ML_Binary32.get_integer_format(),
tag="ik")
twok = ExponentInsertion(ik, precision=self.precision, tag="twok")
retval = Multiplication(poly, twok, tag="retval")
else:
retval = poly
scheme = Return(retval)
return scheme
def generate_expected_table(self, input_tables, table_size_offset_array):
""" Generate the complete table of expected results """
## output values required to check results are stored in output table
num_output_value = self.accuracy.get_num_output_value()
NUM_INPUT_ARRAY = len(input_tables)
TABLE_SIZE = input_tables[0].dimensions[0]
NUM_SUBTABLE = table_size_offset_array.dimensions[0]
EXP_TABLE = [None] * TABLE_SIZE
sum_exp = sollya.SollyaObject(0)
# compute elemnt-wise exponential
for row_id in range(TABLE_SIZE):
local_exp = sollya.exp(input_tables[0][row_id])
EXP_TABLE[row_id] = local_exp
# for each sub-array, compute the sum of exponential, its reciprocal
# and the each element softmax
for table_id in range(NUM_SUBTABLE):
sub_size, sub_offset = table_size_offset_array[table_id]
if sub_size == 0:
# avoid division by zero by skipping empty arrays
continue
sum_exp = sum(EXP_TABLE[sub_offset:(sub_offset + sub_size)])
sum_rcp = 1.0 / sum_exp
for sub_row_id in range(sub_size):
row_id = sub_row_id + sub_offset
EXP_TABLE[row_id] = EXP_TABLE[row_id] * sum_rcp
def expected_value_gen(table_id, table_row_id):
""" generate a full row of expected values using inputs from
input_tables"""
table_offset = table_size_offset_array[table_id][1]
row_id = table_offset + table_row_id
output_values = self.accuracy.get_output_check_value(
EXP_TABLE[row_id])
return output_values
# generating expected value table
expected_table = generate_2d_multi_table(table_size_offset_array,
num_output_value,
self.precision,
"expected_table",
value_gen=expected_value_gen)
return expected_table
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 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
from sollya_extra_functions import cbrt
S2 = sollya.SollyaObject(2)
# dict of (str) -> tuple(ctor, dict(ML_Format -> str))
# the first level key is the function name
# the first value of value tuple is the meta-function constructor
# the second value of the value tuple is a dict which associates to a ML_Format
# the corresponding libm function
FUNCTION_MAP = {
"exp": (metalibm_functions.ml_exp.ML_Exponential, {}, sollya.exp),
"tanh": (metalibm_functions.ml_tanh.ML_HyperbolicTangent, {}, sollya.tanh),
"sqrt": (metalibm_functions.ml_sqrt.MetalibmSqrt, {}, sollya.sqrt),
"log": (metalibm_functions.generic_log.ML_GenericLog, {"basis": sollya.exp(1)}, sollya.log),
"log2": (metalibm_functions.generic_log.ML_GenericLog, {"basis": 2}, sollya.log2),
"log10": (metalibm_functions.generic_log.ML_GenericLog, {"basis": 10}, sollya.log10),
"exp2": (metalibm_functions.ml_exp2.ML_Exp2, {}, (lambda x: S2**x)),
"div": (metalibm_functions.ml_div.ML_Division, {}, (lambda x,y: x / y)),
"cbrt": (metalibm_functions.ml_cbrt.ML_Cbrt, {}, cbrt),
"cosh": (metalibm_functions.ml_cosh.ML_HyperbolicCosine, {}, sollya.cosh),
"sinh": (metalibm_functions.ml_sinh.ML_HyperbolicSine, {}, sollya.sinh),
"cos": (metalibm_functions.ml_sincos.ML_SinCos, {"sin_output": False}, sollya.cos),
"sin": (metalibm_functions.ml_sincos.ML_SinCos, {"sin_output": True}, sollya.sin),
"atan": (metalibm_functions.ml_atan.MetaAtan, {}, sollya.atan),
"atan2": (metalibm_functions.ml_atan.MetaAtan2, {}, (lambda y, x: sollya.atan(y / x))),
class FunctionTest:
def __init__(self, ctor, arg_map_list):
""" FunctionTest constructor:
Args:
ctor(class): constructor of the meta-function
arg_map_list: list of dictionnaries
"""
self.ctor = ctor
self.arg_map_list = arg_map_list
GEN_LOG_ARGS = {
"basis": sollya.exp(1),
"function_name": "ml_genlog",
"passes": ["beforecodegen:fuse_fma"]
}
GEN_LOG2_ARGS = {
"basis": 2,
"function_name": "ml_genlog2",
"passes": ["beforecodegen:fuse_fma"]
}
GEN_LOG10_ARGS = {
"basis": 10,
"function_name": "ml_genlog10",
"passes": ["beforecodegen:fuse_fma"]
}
FUNCTION_LIST = [
from metalibm_core.opt.ml_blocks import (
generate_count_leading_zeros, generate_fasttwosum,
Add222, Add122, Add212, Add211, Mul212, Mul211, Mul222
)
from metalibm_core.opt.p_expand_multi_precision import (
dirty_multi_node_expand
)
from metalibm_core.code_generation.generic_processor import GenericProcessor
from metalibm_core.targets.common.vector_backend import VectorBackend
from metalibm_core.utility.gappa_utils import execute_gappa_script_extract
from metalibm_core.utility.ml_template import *
from metalibm_core.utility.debug_utils import *
EXP_1 = sollya.exp(1)
S2 = sollya.parse("2")
S10 = sollya.parse("10")
def DirtyExponentExtraction(f, precision):
"""Get the raw (biased) exponent of @p f.
Only valid if f is a positive floating-point number.
@param f input node to extract exponent from
@param precision underlying floating-point format
@return node representing f's biased exponent
"""
int_prec = precision.get_integer_format()
field_size = Constant(precision.get_field_size(),
precision=int_prec,
tag='field_size')
recast_f = TypeCast(f, precision=int_prec) \
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
), Return(result_subnormal)), Return(result))))
scheme = pre_scheme
return scheme
def numeric_emulate(self, input_value):
return sollya.log(input_value) / sollya.log(self.basis)
standard_test_cases = [(sollya.parse("0x1.42af3ap-1"), None)]
if __name__ == "__main__":
# auto-test
ARG_TEMPLATE = ML_NewArgTemplate(
default_arg=ML_GenericLog.get_default_args())
ARG_TEMPLATE.parser.add_argument("--basis",
dest="basis",
action="store",
default=exp(1),
type=sollya.parse,
help="logarithm basis")
args = ARG_TEMPLATE.arg_extraction()
ml_log10 = ML_GenericLog(args)
ml_log10.gen_implementation()
def __init__(self,
precision = ML_Binary32,
abs_accuracy = S2**-24,
libm_compliant = True,
debug_flag = False,
fuse_fma = True,
fast_path_extract = True,
target = GenericProcessor(),
output_file = "expf.c",
function_name = "expf"):
# declaring target and instantiating optimization engine
processor = target
self.precision = precision
opt_eng = OptimizationEngine(processor)
gappacg = GappaCodeGenerator(processor, declare_cst = True, disable_debug = True)
# declaring CodeFunction and retrieving input variable
self.function_name = function_name
exp_implementation = CodeFunction(self.function_name, output_format = self.precision)
vx = exp_implementation.add_input_variable("x", self.precision)
Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
test_nan_or_inf = Test(vx, specifier = Test.IsInfOrNaN, likely = False, debug = True, tag = "nan_or_inf")
test_nan = Test(vx, specifier = Test.IsNaN, debug = True, tag = "is_nan_test")
test_positive = Comparison(vx, 0, specifier = Comparison.GreaterOrEqual, debug = True, tag = "inf_sign")
test_signaling_nan = Test(vx, specifier = Test.IsSignalingNaN, debug = True, tag = "is_signaling_nan")
return_snan = Statement(ExpRaiseReturn(ML_FPE_Invalid, return_value = FP_QNaN(self.precision)))
# return in case of infinity input
infty_return = Statement(ConditionBlock(test_positive, Return(FP_PlusInfty(self.precision)), Return(FP_PlusZero(self.precision))))
# return in case of specific value input (NaN or inf)
specific_return = ConditionBlock(test_nan, ConditionBlock(test_signaling_nan, return_snan, Return(FP_QNaN(self.precision))), infty_return)
# return in case of standard (non-special) input
# exclusion of early overflow and underflow cases
precision_emax = self.precision.get_emax()
precision_max_value = S2 * S2**precision_emax
exp_overflow_bound = ceil(log(precision_max_value))
early_overflow_test = Comparison(vx, exp_overflow_bound, likely = False, specifier = Comparison.Greater)
early_overflow_return = Statement(ClearException(), ExpRaiseReturn(ML_FPE_Inexact, ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)))
precision_emin = self.precision.get_emin_subnormal()
precision_min_value = S2 ** precision_emin
exp_underflow_bound = floor(log(precision_min_value))
early_underflow_test = Comparison(vx, exp_underflow_bound, likely = False, specifier = Comparison.Less)
early_underflow_return = Statement(ClearException(), ExpRaiseReturn(ML_FPE_Inexact, ML_FPE_Underflow, return_value = FP_PlusZero(self.precision)))
sollya_prec_map = {ML_Binary32: sollya.binary32, ML_Binary64: sollya.binary64}
# constant computation
invlog2 = round(1/log(2), sollya_prec_map[self.precision], RN)
interval_vx = Interval(exp_underflow_bound, exp_overflow_bound)
interval_fk = interval_vx * invlog2
interval_k = Interval(floor(inf(interval_fk)), ceil(sup(interval_fk)))
log2_hi_precision = self.precision.get_field_size() - (ceil(log2(sup(abs(interval_k)))) + 2)
Log.report(Log.Info, "log2_hi_precision: "), log2_hi_precision
invlog2_cst = Constant(invlog2, precision = self.precision)
log2_hi = round(log(2), log2_hi_precision, sollya.RN)
log2_lo = round(log(2) - log2_hi, sollya_prec_map[self.precision], sollya.RN)
# argument reduction
unround_k = vx * invlog2
unround_k.set_attributes(tag = "unround_k", debug = ML_Debug(display_format = "%f"))
k = NearestInteger(unround_k, precision = self.precision, debug = ML_Debug(display_format = "%f"))
ik = NearestInteger(unround_k, precision = ML_Int32, debug = ML_Debug(display_format = "%d"), tag = "ik")
ik.set_tag("ik")
k.set_tag("k")
exact_pre_mul = (k * log2_hi)
exact_pre_mul.set_attributes(exact= True)
exact_hi_part = vx - exact_pre_mul
exact_hi_part.set_attributes(exact = True)
r = exact_hi_part - k * log2_lo
r.set_tag("r")
r.set_attributes(debug = ML_Debug(display_format = "%f"))
opt_r = opt_eng.optimization_process(r, self.precision, copy = True, fuse_fma = fuse_fma)
tag_map = {}
opt_eng.register_nodes_by_tag(opt_r, tag_map)
cg_eval_error_copy_map = {
vx: Variable("x", precision = self.precision, interval = interval_vx),
tag_map["k"]: Variable("k", interval = interval_k, precision = self.precision)
}
#try:
if 1:
#eval_error = gappacg.get_eval_error(opt_r, cg_eval_error_copy_map, gappa_filename = "red_arg.g")
eval_error = gappacg.get_eval_error_v2(opt_eng, opt_r, cg_eval_error_copy_map, gappa_filename = "red_arg.g")
Log.report(Log.Info, "eval error: %s" % eval_error)
#except:
# Log.report(Log.Info, "gappa error evaluation failed")
print r.get_str(depth = None, display_precision = True, display_attribute = True)
print opt_r.get_str(depth = None, display_precision = True, display_attribute = True)
approx_interval = Interval(-log(2)/2, log(2)/2)
local_ulp = sup(ulp(exp(approx_interval), self.precision))
print "ulp: ", local_ulp
error_goal = local_ulp #S2**-(self.precision.get_field_size()+1)
error_goal_approx = S2**-1 * error_goal
Log.report(Log.Info, "\033[33;1m building mathematical polynomial \033[0m\n")
poly_degree = sup(guessdegree(exp(x), approx_interval, error_goal_approx)) #- 1
init_poly_degree = poly_degree
return
while 1:
Log.report(Log.Info, "attempting poly degree: %d" % poly_degree)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(exp(x), poly_degree, [self.precision]*(poly_degree+1), approx_interval, absolute)
Log.report(Log.Info, "poly approx error: %s" % poly_approx_error)
Log.report(Log.Info, "\033[33;1m generating polynomial evaluation scheme \033[0m")
poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, r, unified_precision = self.precision)
poly.set_tag("poly")
# optimizing poly before evaluation error computation
opt_poly = opt_eng.optimization_process(poly, self.precision)
#print "poly: ", poly.get_str(depth = None, display_precision = True)
#print "opt_poly: ", opt_poly.get_str(depth = None, display_precision = True)
# evaluating error of the polynomial approximation
r_gappa_var = Variable("r", precision = self.precision, interval = approx_interval)
poly_error_copy_map = {
r.get_handle().get_node(): r_gappa_var
}
gappacg = GappaCodeGenerator(target, declare_cst = False, disable_debug = True)
poly_eval_error = gappacg.get_eval_error_v2(opt_eng, poly.get_handle().get_node(), poly_error_copy_map, gappa_filename = "gappa_poly.g")
Log.report(Log.Info, "poly evaluation error: %s" % poly_eval_error)
global_poly_error = poly_eval_error + poly_approx_error
global_rel_poly_error = global_poly_error / exp(approx_interval)
print "global_poly_error: ", global_poly_error, global_rel_poly_error
flag = local_ulp > sup(abs(global_rel_poly_error))
print "test: ", flag
if flag: break
else:
if poly_degree > init_poly_degree + 5:
Log.report(Log.Error, "poly degree search did not converge")
poly_degree += 1
late_overflow_test = Comparison(ik, self.precision.get_emax(), specifier = Comparison.Greater, likely = False, debug = True, tag = "late_overflow_test")
overflow_exp_offset = (self.precision.get_emax() - self.precision.get_field_size() / 2)
diff_k = ik - overflow_exp_offset
diff_k.set_attributes(debug = ML_Debug(display_format = "%d"), tag = "diff_k")
late_overflow_result = (ExponentInsertion(diff_k) * poly) * ExponentInsertion(overflow_exp_offset)
late_overflow_result.set_attributes(silent = False, tag = "late_overflow_result", debug = debugf)
late_overflow_return = ConditionBlock(Test(late_overflow_result, specifier = Test.IsInfty, likely = False), ExpRaiseReturn(ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)), Return(late_overflow_result))
late_underflow_test = Comparison(k, self.precision.get_emin_normal(), specifier = Comparison.LessOrEqual, likely = False)
underflow_exp_offset = 2 * self.precision.get_field_size()
late_underflow_result = (ExponentInsertion(ik + underflow_exp_offset) * poly) * ExponentInsertion(-underflow_exp_offset)
late_underflow_result.set_attributes(debug = ML_Debug(display_format = "%e"), tag = "late_underflow_result", silent = False)
test_subnormal = Test(late_underflow_result, specifier = Test.IsSubnormal)
late_underflow_return = Statement(ConditionBlock(test_subnormal, ExpRaiseReturn(ML_FPE_Underflow, return_value = late_underflow_result)), Return(late_underflow_result))
std_result = poly * ExponentInsertion(ik, tag = "exp_ik", debug = debug_lftolx)
std_result.set_attributes(tag = "std_result", debug = debug_lftolx)
result_scheme = ConditionBlock(late_overflow_test, late_overflow_return, ConditionBlock(late_underflow_test, late_underflow_return, Return(std_result)))
std_return = ConditionBlock(early_overflow_test, early_overflow_return, ConditionBlock(early_underflow_test, early_underflow_return, result_scheme))
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = ConditionBlock(test_nan_or_inf, Statement(ClearException(), specific_return), std_return)
#print scheme.get_str(depth = None, display_precision = True)
# fusing FMA
if fuse_fma:
Log.report(Log.Info, "\033[33;1m MDL fusing FMA \033[0m")
scheme = opt_eng.fuse_multiply_add(scheme, silence = True)
Log.report(Log.Info, "\033[33;1m MDL abstract scheme \033[0m")
opt_eng.instantiate_abstract_precision(scheme, None)
Log.report(Log.Info, "\033[33;1m MDL instantiated scheme \033[0m")
opt_eng.instantiate_precision(scheme, default_precision = self.precision)
Log.report(Log.Info, "\033[33;1m subexpression sharing \033[0m")
opt_eng.subexpression_sharing(scheme)
Log.report(Log.Info, "\033[33;1m silencing operation \033[0m")
opt_eng.silence_fp_operations(scheme)
# registering scheme as function implementation
exp_implementation.set_scheme(scheme)
# check processor support
Log.report(Log.Info, "\033[33;1m checking processor support \033[0m")
opt_eng.check_processor_support(scheme)
# factorizing fast path
if fast_path_extract:
Log.report(Log.Info, "\033[33;1m factorizing fast path\033[0m")
opt_eng.factorize_fast_path(scheme)
Log.report(Log.Info, "\033[33;1m generating source code \033[0m")
cg = CCodeGenerator(processor, declare_cst = False, disable_debug = not debug_flag, libm_compliant = libm_compliant)
self.result = exp_implementation.get_definition(cg, C_Code, static_cst = True)
#self.result.add_header("support_lib/ml_types.h")
self.result.add_header("support_lib/ml_special_values.h")
self.result.add_header_comment("polynomial degree for exp(x): %d" % poly_degree)
self.result.add_header_comment("sollya polynomial for exp(x): %s" % poly_object.get_sollya_object())
if debug_flag:
self.result.add_header("stdio.h")
self.result.add_header("inttypes.h")
output_stream = open(output_file, "w")#"%s.c" % exp_implementation.get_name(), "w")
output_stream.write(self.result.get(cg))
output_stream.close()
Log.report(Log.Error, "poly degree must be positive or null. {}, {}",
poly_object.degree, poly_object)
if __name__ == "__main__":
implem_results = []
for eps_target in [S2**-40, S2**-50, S2**-55, S2**-60, S2**-65]:
approx_interval = Interval(-S2**-5, S2**-5)
ctx = MLL_Context(ML_Binary64, approx_interval)
vx = Variable("x",
precision=ctx.variableFormat,
interval=approx_interval)
# guessding the best degree
poly_degree = int(
sup(
sollya.guessdegree(sollya.exp(sollya.x), approx_interval,
eps_target)))
# asking sollya to provide the approximation
poly_object = Polynomial.build_from_approximation(
sollya.exp(sollya.x), poly_degree,
[sollya.doubledouble] * (poly_degree + 1), vx.interval)
print("poly object is {}".format(poly_object))
poly_graph, poly_epsilon = mll_implementpoly_horner(
ctx, poly_object, eps_target, vx)
print("poly_graph is {}".format(
poly_graph.get_str(depth=None, display_precision=True)))
print("poly epsilon is {}".format(float(poly_epsilon)))
print("poly accuracy is {}".format(
get_accuracy_from_epsilon(poly_epsilon)))
implem_results.append(
(eps_target, poly_degree, poly_object, poly_graph, poly_epsilon))
ctor(class): constructor of the meta-function
arg_map_list: list of dictionnaries
"""
self.ctor = ctor
self.arg_map_list = arg_map_list
self.title = title if not title is None else ctor.function_name
# callback(<option dict>) -> specialized <option dict>
self.specific_opts_builder = specific_opts_builder
@property
def tag(self):
return self.ctor.function_name
GEN_LOG_ARGS = {"basis": sollya.exp(1), "function_name": "ml_genlog", "extra_passes" : ["beforecodegen:fuse_fma"]}
GEN_LOG2_ARGS = {"basis": 2, "function_name": "ml_genlog2", "extra_passes" : ["beforecodegen:fuse_fma"]}
GEN_LOG10_ARGS = {"basis": 10, "function_name": "ml_genlog10", "extra_passes" : ["beforecodegen:fuse_fma"]}
class LibmFunctionTest(FunctionTest):
@property
def tag(self):
# NOTES/FIXME: 0-th element of self.arg_map_list is chosen
# for tag determination without considering the others
return self.title + "_" + self.arg_map_list[0]["bench_function_name"]
S2 = sollya.SollyaObject(2)
S10 = sollya.SollyaObject(10)
def emulate_exp2(v):
return S2**v
def emulate_exp10(v):
def generate_reduction_fptaylor(x):
# get k, must be the same at endpoints
unround_k = x * n_invlog2
k_low = sollya.floor(sollya.inf(unround_k))
k_high = sollya.floor(sollya.sup(unround_k))
if not (k_low == k_high or (k_low == -1 and sollya.sup(x) == 0)):
assert False, "Interval must not straddle multples of log(2)"
k = int(k_low)
r = x - k * n_log2
twok = 2**k
x_low = sollya.inf(x)
x_high = sollya.sup(x)
query = "\n".join([
"Variables", " real x in [{},{}];".format(x_low, x_high),
"Definitions", " whole rnd64= {} * {};".format(k, n_log2),
" r rnd64= x - whole;", " poly rnd64= {};".format(poly_expr),
" retval rnd64= poly*{};".format(twok), "Expressions",
" retval;"
])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "true",
"--abs-error": "true"
})
rnd_rel_err = float(
res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "false",
"--abs-error": "true"
})
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.exp(sollya.x), r, sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.exp(sollya.x), r, sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def numeric_emulate(self, input_value):
""" Numeric emaluation of exponential """
return sollya.exp(input_value)
def numeric_emulate(self, input_value):
return exp(input_value)
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 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
