def standard_test_cases(self):
fp64_list = [
# random test
(sollya.parse("0x1.e906cc97d7cc1p+743"), sollya.parse("0x0.000001b84ba98p-1022")),
(sollya.parse("0x1.9c4110b0dea4fp+279"), sollya.parse("0x0.000ccf2945bd8p-1022")),
# OpenCL CTS error
# infinite loop
# ERROR: remquoD: {-inf, 77} ulp error at {0x0.eaffffffffb86p-1022, -0x0.0000000000202p-1022} ({ 0x000eaffffffffb86, 0x8000000000000202}): *{0x0.00000000000eap-1022, -78} ({ 0x00000000000000ea, 0xffffffb2}) vs. {-0x1.0000000000000p+0, -1} ({ 0xbff0000000000000, 0xffffffff})
(sollya.parse("0x0.eaffffffffb86p-1022"), sollya.parse("-0x0.0000000000202p-1022"), sollya.parse("0x0.00000000000eap-1022") if self.mode is REMAINDER_MODE else -78),
# ERROR: remquoD: {0.000000, 92} ulp error at {-0x1.b9000000003e0p-982, -0x0.0000000000232p-1022} ({ 0x829b9000000003e0, 0x8000000000000232}): *{-0x0.0000000000000p+0, 0} ({ 0x8000000000000000, 0x00000000}) vs. {-0x0.0000000000000p+0, 92} ({ 0x8000000000000000, 0x0000005c})
(sollya.parse("-0x1.b9000000003e0p-982"), sollya.parse("-0x0.0000000000232p-1022"), FP_MinusZero(self.precision) if self.mode is REMAINDER_MODE else 0),
# ERROR: remquoD: {-26458647810801664.000000, 1} ulp error at {-0x1.be000000005dfp+977, 0x1.78000000006f1p+975} ({ 0xfd0be000000005df, 0x7ce78000000006f1}): *{0x1.8000000002ce4p+973, -5} ({ 0x7cc8000000002ce4, 0xfffffffb}) vs. {-0x1.17ffffffffbb8p+975, -4} ({ 0xfce17ffffffffbb8, 0xfffffffc})
(sollya.parse("-0x1.be000000005dfp+977"), sollya.parse("0x1.78000000006f1p+975"), sollya.parse("0x1.8000000002ce4p+973") if self.mode is REMAINDER_MODE else -5),
# ERROR: remquoD: {-171.000000, -1} ulp error at {-0x1.02ffffffffc2bp+489, -0x0.00000000000abp-1022} ({ 0xde802ffffffffc2b, 0x80000000000000ab}): *{0x0.0000000000055p-1022, 127} ({ 0x0000000000000055, 0x0000007f}) vs. {-0x0.0000000000056p-1022, 254} ({ 0x8000000000000056, 0x000000fe})
(sollya.parse("0x1.02ffffffffc2bp+489"), sollya.parse("0x0.00000000000abp-1022")),
(sollya.parse("-0x1.02ffffffffc2bp+489"), sollya.parse("-0x0.00000000000abp-1022"), sollya.parse("0x0.0000000000055p-1022") if self.mode is REMAINDER_MODE else 127),
# ERROR: remquoD: {nan, 0} ulp error at {-0x1.69000000001e1p+749, -inf} ({ 0xeec69000000001e1, 0xfff0000000000000}): *{-0x1.69000000001e1p+749, 0} ({ 0xeec69000000001e1, 0x00000000}) vs. {nan, 0} ({ 0x7ff8000000000000, 0x00000000})
(sollya.parse("-0x1.69000000001e1p+749"), FP_MinusInfty(self.precision), sollya.parse("-0x1.69000000001e1p+749") if self.mode is REMAINDER_MODE else 0),
# ERROR: remquo: {0.000000, -126} ulp error at {0x1.921456p+70, -0x1.921456p+70} ({0x62c90a2b, 0xe2c90a2b}): *{0x0p+0, -1} ({0x00000000, 0xffffffff}) vs. {0x0p+0, 1} ({0x00000000, 0x00000001})
(sollya.parse("0x1.921456p+70"), sollya.parse("-0x1.921456p+70"), 0 if self.mode is REMAINDER_MODE else -1),
# ERROR: remquoD: {10731233487093760.000000, 0} ulp error at {0x1.30fffffffff7ep-597, -0x1.13ffffffffed0p-596} ({ 0x1aa30fffffffff7e, 0x9ab13ffffffffed0}): *{-0x1.edffffffffc44p-598, -1} ({ 0x9a9edffffffffc44, 0xffffffff}) vs. {0x1.d000000000ae0p-600, -1} ({ 0x1a7d000000000ae0, 0xffffffff})
(sollya.parse("0x1.30fffffffff7ep-597"), sollya.parse("-0x1.13ffffffffed0p-596"), -1 if self.mode is QUOTIENT_MODE else sollya.parse("-0x1.edffffffffc44p-598")),
# {-0x1.9bffffffffd38p+361, 0x0.00000000000a5p-1022} ({ 0xd689bffffffffd38, 0x00000000000000a5}): *{-0x0.000000000000fp-1022, -93} ({ 0x800000000000000f, 0xffffffa3}) vs. {0x0.000000000000fp-1022, -1171354717} ({ 0x000000000000000f, 0xba2e8ba3})
(sollya.parse("-0x1.9bffffffffd38p+361"), sollya.parse("0x0.00000000000a5p-1022")),
(sollya.parse("0x0.ac0f94b9da13p-1022"), sollya.parse("-0x1.1e4580d7eb2e7p-1022")),
(sollya.parse("0x1.fffffffffffffp+1023"), sollya.parse("-0x1.fffffffffffffp+1023")),
(sollya.parse("0x0.a9f466178b1fcp-1022"), sollya.parse("0x0.b22f552dc829ap-1022")),
(sollya.parse("0x1.4af8b07942537p-430"), sollya.parse("-0x0.f72be041645b7p-1022")),
#result is 0x0.0000000000505p-1022 vs expected0x0.0000000000a3cp-1022
#(sollya.parse("0x1.9f9f4e9a29fcfp-421"), sollya.parse("0x0.0000000001b59p-1022"), sollya.parse("0x0.0000000000a3cp-1022")),
(sollya.parse("0x1.9906165fb3e61p+62"), sollya.parse("0x1.9906165fb3e61p+60")),
(sollya.parse("0x1.9906165fb3e61p+62"), sollya.parse("0x0.0000000005e7dp-1022")),
(sollya.parse("0x1.77f00143ba3f4p+943"), sollya.parse("0x0.000000000001p-1022")),
(sollya.parse("0x0.000000000001p-1022"), sollya.parse("0x0.000000000001p-1022")),
(sollya.parse("0x0.000000000348bp-1022"), sollya.parse("0x0.000000000001p-1022")),
(sollya.parse("0x1.bcf3955c3b244p-130"), sollya.parse("0x1.77aef33890951p-1003")),
(sollya.parse("0x1.8de59bd84c51ep-866"), sollya.parse("0x1.045aa9bf14fb1p-774")),
(sollya.parse("0x1.9f9f4e9a29fcfp-421"), sollya.parse("0x0.0000000001b59p-1022")),
(sollya.parse("0x1.2e1c59b43a459p+953"), sollya.parse("0x0.0000001cf5319p-1022")),
(sollya.parse("-0x1.86c83abe0854ep+268"), FP_MinusInfty(self.precision)),
# bad sign of remainder
(sollya.parse("0x1.d3fb9968850a5p-960"), sollya.parse("-0x0.23c1ed19c45fp-1022")),
# bad sign of zero
(FP_MinusZero(self.precision), sollya.parse("0x1.85200a9235193p-450")),
# bad remainder
(sollya.parse("0x1.fffffffffffffp+1023"), sollya.parse("0x1.1f31bcd002a7ap-803")),
# bad sign
(sollya.parse("-0x1.4607d0c9fc1a7p-878"), sollya.parse("-0x1.9b666b840b1bp-1023")),
]
return fp64_list if self.precision.get_bit_size() >= 64 else []
def __call__(args):
PRECISION = ML_Binary64
value_list = [
FP_PlusInfty(PRECISION),
FP_MinusInfty(PRECISION),
FP_PlusZero(PRECISION),
FP_MinusZero(PRECISION),
FP_QNaN(PRECISION),
FP_SNaN(PRECISION),
#FP_PlusOmega(PRECISION),
#FP_MinusOmega(PRECISION),
NumericValue(7.0),
NumericValue(-3.0),
]
op_map = {
"+": operator.__add__,
"-": operator.__sub__,
"*": operator.__mul__,
}
for op in op_map:
for lhs in value_list:
for rhs in value_list:
print("{} {} {} = {}".format(lhs, op, rhs,
op_map[op](lhs, rhs)))
return True
def get_integer_coding(self, value, language=C_Code):
if FP_SpecialValue.is_special_value(value):
return self.get_special_value_coding(value, language)
elif value == ml_infty:
return self.get_special_value_coding(FP_PlusInfty(self), language)
elif value == -ml_infty:
return self.get_special_value_coding(FP_MinusInfty(self), language)
else:
value = sollya.round(value, self.get_sollya_object(), sollya.RN)
# FIXME: managing negative zero
sign = int(1 if value < 0 else 0)
value = abs(value)
if value == 0.0:
Log.report(Log.Warning,
"+0.0 forced during get_integer_coding conversion")
exp_biased = 0
mant = 0
else:
exp = int(sollya.floor(sollya.log2(value)))
exp_biased = int(exp - self.get_bias())
if exp < self.get_emin_normal():
exp_biased = 0
mant = int((value / S2**self.get_emin_subnormal()))
else:
mant = int(
(value / S2**exp - 1.0) * (S2**self.get_field_size()))
return mant | (exp_biased << self.get_field_size()) | (
sign << (self.get_field_size() + self.get_exponent_size()))
def __call__(args):
for PRECISION in [ML_Binary32, ML_Binary64]:
TEST_CASE = [
(FP_PlusOmega(PRECISION), "<", FP_SNaN(PRECISION), Unordered),
(FP_PlusOmega(PRECISION), "<=", FP_SNaN(PRECISION), Unordered),
(FP_PlusInfty(PRECISION), "==", FP_PlusInfty(PRECISION), True),
(FP_PlusInfty(PRECISION), "==", FP_QNaN(PRECISION), False),
(FP_PlusZero(PRECISION), "==", FP_MinusZero(PRECISION), True),
(FP_PlusZero(PRECISION), "!=", FP_MinusZero(PRECISION), False),
(FP_QNaN(PRECISION), "==", FP_QNaN(PRECISION), False),
(FP_PlusInfty(PRECISION), "==", -FP_MinusInfty(PRECISION), True),
(FP_MinusInfty(PRECISION), ">", 0, False),
(FP_MinusInfty(PRECISION), ">", FP_PlusZero(PRECISION), False),
(FP_MinusInfty(PRECISION), ">", FP_MinusZero(PRECISION), False),
]
for lhs, op, rhs, expected in TEST_CASE:
result = op_map[op](lhs, rhs)
assert result == expected, "failure: {} {} {} = {} vs expected {} ".format(lhs, op, rhsresult, expected)
return True
def numeric_emulate(self, vx, n):
""" Numeric emulation of n-th root """
if FP_SpecialValue.is_special_value(vx):
if is_nan(vx):
return FP_QNaN(self.precision)
elif is_plus_infty(vx):
return SOLLYA_INFTY
elif is_minus_infty(vx):
if int(n) % 2 == 1:
return vx
else:
return FP_QNaN(self.precision)
elif is_zero(vx):
if int(n) % 2 != 0 and n < 0:
if is_plus_zero(vx):
return FP_PlusInfty(self.precision)
else:
return FP_MinusInfty(self.precision)
elif int(n) % 2 == 0:
if n < 0:
return FP_PlusInfty(self.precision)
elif n > 0:
return FP_PlusZero(self.precision)
return FP_QNaN(self.precision)
else:
raise NotImplementedError
# OpenCL-C rootn, x < 0 and y odd: -exp2(log2(-x) / y)
S2 = sollya.SollyaObject(2)
if vx < 0:
if int(n) % 2 != 0:
if n > 0:
v = -bigfloat.root(
sollya.SollyaObject(-vx).bigfloat(), int(n))
else:
v = -S2**(sollya.log2(-vx) / n)
else:
return FP_QNaN(self.precision)
elif n < 0:
# OpenCL-C definition
v = S2**(sollya.log2(vx) / n)
else:
v = bigfloat.root(sollya.SollyaObject(vx).bigfloat(), int(n))
return sollya.SollyaObject(v)
def generate_scheme(self):
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
sollya_precision = self.get_input_precision().get_sollya_object()
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# testing special value inputs
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
# if input is a signaling NaN, raise an invalid exception and returns
# a quiet NaN
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = self.precision.get_integer_format()
# log2(vx)
# r = vx_mant
# e = vx_exp
# vx reduced to r in [1, 2[
# log2(vx) = log2(r * 2^e)
# = log2(r) + e
#
## log2(r) is approximated by
# log2(r) = log2(inv_seed(r) * r / inv_seed(r)
# = log2(inv_seed(r) * r) - log2(inv_seed(r))
# inv_seed(r) in ]1/2, 1] => log2(inv_seed(r)) in ]-1, 0]
#
# inv_seed(r) * r ~ 1
# we can easily tabulate -log2(inv_seed(r))
#
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("inv_table"))
# value for index 0 is set to 0.0
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in range(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
#inv_value = (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
#print inv_approx_table[i][0], inv_value
inv_value = inv_approx_table[i][0]
value_high_bitsize = self.precision.get_field_size() - (
self.precision.get_exponent_size() + 1)
value_high = round(log2(inv_value), value_high_bitsize, sollya.RN)
value_low = round(
log2(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_lftolx)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debugd)
# The main table is indexed by the 7 most significant bits
# of the mantissa
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debuglld)
# argument reduction
# Using AND -2 to exclude LSB set to 1 for Newton-Raphson convergence
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(DivisionSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_lftolx,
silent=True),
precision=ML_UInt64),
Constant(-2, precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
_red_vx = FMA(arg_red_index, _vx_mant, -1.0)
_red_vx.set_attributes(tag="_red_vx", debug=debug_lftolx)
inv_err = S2**-inv_approx_table.index_size
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log2(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() * 1.1))) + 1
sollya.settings.display = sollya.hexadecimal
global_poly_object, approx_error = Polynomial.build_from_approximation_with_error(
log2(1 + sollya.x) / sollya.x,
poly_degree, [self.precision] * (poly_degree + 1),
approx_interval,
sollya.absolute,
error_function=lambda p, f, ai, mod, t: sollya.dirtyinfnorm(
p - f, ai))
Log.report(
Log.Info, "poly_degree={}, approx_error={}".format(
poly_degree, approx_error))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
#poly_object = global_poly_object.sub_poly(start_index=0,offset=0)
Attributes.set_default_silent(True)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly = FMA(pre_poly, _red_vx,
global_poly_object.get_cst_coeff(0, self.precision))
_poly.set_attributes(tag="poly", debug=debug_lftolx)
Log.report(
Log.Verbose, "sollya global_poly_object: {}".format(
global_poly_object.get_sollya_object()))
Log.report(
Log.Verbose, "sollya poly_object: {}".format(
poly_object.get_sollya_object()))
corr_exp = _vx_exp if exp_corr_factor == None else _vx_exp + exp_corr_factor
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
pre_result = -_log_inv_hi + (_red_vx * _poly + (-_log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = Conversion(corr_exp, precision=self.precision)
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_hex",
debug=debug_lftolx)
_result = exact_log2_hi_exp + pre_result
return _result, _poly, _log_inv_lo, _log_inv_hi, _red_vx
result, poly, log_inv_lo, log_inv_hi, red_vx = compute_log(vx)
result.set_attributes(tag="result", debug=debug_lftolx)
# specific input value predicate
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="vx_snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="vx_inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debugd,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debugd,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debugd)
# Specific specific for the case exp == -1
# log2(x) = log2(m) - 1
#
# as m in [1, 2[, log2(m) in [0, 1[
# if r is close to 2, a catastrophic cancellation can occur
#
# r = seed(m)
# log2(x) = log2(seed(m) * m / seed(m)) - 1
# = log2(seed(m) * m) - log2(seed(m)) - 1
#
# for m really close to 2 => seed(m) = 0.5
# => log2(x) = log2(0.5 * m)
# =
result_exp_m1 = (-log_inv_hi - 1.0) + FMA(poly, red_vx, -log_inv_lo)
result_exp_m1.set_attributes(tag="result_exp_m1", debug=debug_lftolx)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
result_subnormal.set_attributes(tag="result_subnormal",
debug=debug_lftolx)
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log(1 + x) / x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_lftolx)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one", debug=debugd, likely=False)
# main scheme
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
),
Statement(ClearException(), result_subnormal,
Return(result_subnormal))),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result_exp_m1),
Return(result))))))
scheme = Statement(result, pre_scheme)
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.precision.sollya_object
# constant computation
invlog2 = round(1 / log(2), sollya_precision, sollya.RN)
invlog2_cst = Constant(invlog2, precision=self.precision)
#v_log2_hi = round(log(2), 16, sollya.RN)
#v_log2_lo = round(log(2) - v_log2_hi, sollya_precision, sollya.RN)
#log2_hi = Constant(v_log2_hi, precision = self.precision, tag = "log2_hi")
#log2_lo = Constant(v_log2_lo, precision = self.precision, tag = "log2_lo")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
v_log2_hi = round(
log(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
v_log2_lo = round(
log(2) - v_log2_hi, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(v_log2_hi, precision=self.precision, tag="log2_hi")
log2_lo = Constant(v_log2_lo, precision=self.precision, tag="log2_lo")
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debug_multi)
int_precision = self.precision.get_integer_format()
# table creation
table_index_size = 7
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("inv_table"))
log_table[0][0] = 0.0
log_table[0][1] = 0.0
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
integer_precision = {
ML_Binary32: ML_UInt32,
ML_Binary64: ML_UInt64
}[self.precision]
for i in range(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
inv_value = inv_approx_table[
i] # (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
value_high = round(
log(inv_value),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(
log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
debug=debug_multi,
precision=self.precision)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debug_multi)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(_vx_mant, precision=int_precision, debug=debug_multi),
self.precision.get_field_size() - 7,
debug=debug_multi),
0x7f,
tag="table_index",
debug=debug_multi)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=integer_precision),
Constant(-2, precision=integer_precision),
precision=integer_precision),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
arg_red_index = Select(Equal(table_index, 0), 1.0,
pre_arg_red_index)
#_red_vx = arg_red_index * _vx_mant - 1.0
_red_vx = FusedMultiplyAdd(arg_red_index,
_vx_mant,
1.0,
specifier=FusedMultiplyAdd.Subtract)
_red_vx.set_attributes(tag="_red_vx", debug=debug_multi)
inv_err = S2**-7
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_multi)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree,
[1] + [self.precision] * (poly_degree), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
_poly = PolynomialSchemeEvaluator.generate_estrin_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
corr_exp = Conversion(
_vx_exp if exp_corr_factor == None else _vx_exp +
exp_corr_factor,
precision=self.precision)
split_red_vx = Split(_red_vx,
precision=ML_DoubleDouble,
tag="split_red_vx",
debug=debug_multi)
red_vx_hi = split_red_vx.hi
red_vx_lo = split_red_vx.lo
# result = _red_vx * poly - log_inv_hi - log_inv_lo + _vx_exp * log2_hi + _vx_exp * log2_lo
pre_result = -_log_inv_hi + (_red_vx +
(_red_vx * _poly +
(corr_exp * log2_lo - _log_inv_lo)))
pre_result.set_attributes(tag="pre_result", debug=debug_multi)
exact_log2_hi_exp = corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_exp",
debug=debug_multi)
cancel_part = (corr_exp * log2_hi - _log_inv_hi)
cancel_part.set_attributes(tag="cancel_part", debug=debug_multi)
sub_part = red_vx_hi + cancel_part
sub_part.set_attributes(tag="sub_part", debug=debug_multi)
#result_one_low_part = (red_vx_hi * _poly + (red_vx_lo + (red_vx_lo * _poly + (corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part = ((red_vx_lo +
(red_vx_lo * _poly +
(corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part.set_attributes(tag="result_one_low_part",
debug=debug_multi)
_result_one = (
(sub_part) + red_vx_hi * _poly) + result_one_low_part
return exact_log2_hi_exp + pre_result, _poly, _log_inv_lo, _log_inv_hi, _red_vx, _result_one
result, poly, log_inv_lo, log_inv_hi, red_vx, new_result_one = compute_log(
vx)
result.set_attributes(tag="result", debug=debug_multi)
new_result_one.set_attributes(tag="new_result_one", debug=debug_multi)
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debug_multi,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debug_multi,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debug_multi,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debug_multi,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debug_multi,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debug_multi,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debug_multi)
# exp=-1 case
Log.report(Log.Verbose, "managing exp=-1 case")
result2 = (-log_inv_hi - log2_hi) + (
(red_vx + poly * red_vx) - log2_lo - log_inv_lo)
result2.set_attributes(tag="result2", debug=debug_multi)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
Log.report(Log.Verbose, "managing close to 1.0 cases")
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
sollya.absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_multi)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one",
debug=debug_multi,
likely=False)
# main scheme
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
), Return(result_subnormal)),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result2),
Return(result))
#ConditionBlock(cond_one,
#Return(new_result_one),
#ConditionBlock(exp_mone,
#Return(result2),
#Return(result)
#)
#)
))))
scheme = pre_scheme
return scheme
def generate_scalar_scheme(self, vx, vy):
# fixing inputs' node tag
vx.set_attributes(tag="x")
vy.set_attributes(tag="y")
int_precision = self.precision.get_integer_format()
# assuming x = m.2^e (m in [1, 2[)
# n, positive or null integers
#
# pow(x, n) = x^(y)
# = exp(y * log(x))
# = 2^(y * log2(x))
# = 2^(y * (log2(m) + e))
#
e = ExponentExtraction(vx, tag="e", precision=int_precision)
m = MantissaExtraction(vx, tag="m", precision=self.precision)
# approximation log2(m)
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision = self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision = self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed, language=None,
table_getter= lambda self: self.approx_table_map)
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
ml_log_args = ML_GenericLog.get_default_args(precision=self.precision, basis=2)
ml_log = ML_GenericLog(ml_log_args)
log_table, log_table_tho, table_index_range = ml_log.generate_log_table(log_f, inv_approx_table)
log_approx = ml_log.generate_reduced_log_split(Abs(m, precision=self.precision), log_f, inv_approx_table, log_table)
log_approx = Select(Equal(vx, 0), FP_MinusInfty(self.precision), log_approx)
log_approx.set_attributes(tag="log_approx", debug=debug_multi)
r = Multiplication(log_approx, vy, tag="r", debug=debug_multi)
# 2^(y * (log2(m) + e)) = 2^(y * log2(m)) * 2^(y * e)
#
# log_approx = log2(Abs(m))
# r = y * log_approx ~ y * log2(m)
#
# NOTES: manage cases where e is negative and
# (y * log2(m)) AND (y * e) could cancel out
# if e positive, whichever the sign of y (y * log2(m)) and (y * e) CANNOT
# be of opposite signs
# log2(m) in [0, 1[ so cancellation can occur only if e == -1
# we split 2^x in 2^x = 2^t0 * 2^t1
# if e < 0: t0 = y * (log2(m) + e), t1=0
# else: t0 = y * log2(m), t1 = y * e
t_cond = e < 0
# e_y ~ e * y
e_f = Conversion(e, precision=self.precision)
#t0 = Select(t_cond, (e_f + log_approx) * vy, Multiplication(e_f, vy), tag="t0")
#NearestInteger(t0, precision=self.precision, tag="t0_int")
EY = NearestInteger(e_f * vy, tag="EY", precision=self.precision)
LY = NearestInteger(log_approx * vy, tag="LY", precision=self.precision)
t0_int = Select(t_cond, EY + LY, EY, tag="t0_int")
t0_frac = Select(t_cond, FMA(e_f, vy, -EY) + FMA(log_approx, vy, -LY) ,EY - t0_int, tag="t0_frac")
#t0_frac.set_attributes(tag="t0_frac")
ml_exp2_args = ML_Exp2.get_default_args(precision=self.precision)
ml_exp2 = ML_Exp2(ml_exp2_args)
exp2_t0_frac = ml_exp2.generate_scalar_scheme(t0_frac, inline_select=True)
exp2_t0_frac.set_attributes(tag="exp2_t0_frac", debug=debug_multi)
exp2_t0_int = ExponentInsertion(Conversion(t0_int, precision=int_precision), precision=self.precision, tag="exp2_t0_int")
t1 = Select(t_cond, Constant(0, precision=self.precision), r)
exp2_t1 = ml_exp2.generate_scalar_scheme(t1, inline_select=True)
exp2_t1.set_attributes(tag="exp2_t1", debug=debug_multi)
result_sign = Constant(1.0, precision=self.precision) # Select(n_is_odd, CopySign(vx, Constant(1.0, precision=self.precision)), 1)
y_int = NearestInteger(vy, precision=self.precision)
y_is_integer = Equal(y_int, vy)
y_is_even = LogicalOr(
# if y is a number (exc. inf) greater than 2**mantissa_size * 2,
# then it is an integer multiple of 2 => even
Abs(vy) >= 2**(self.precision.get_mantissa_size()+1),
LogicalAnd(
y_is_integer and Abs(vy) < 2**(self.precision.get_mantissa_size()+1),
# we want to limit the modulo computation to an integer input
Equal(Modulo(Conversion(y_int, precision=int_precision), 2), 0)
)
)
y_is_odd = LogicalAnd(
LogicalAnd(
Abs(vy) < 2**(self.precision.get_mantissa_size()+1),
y_is_integer
),
Equal(Modulo(Conversion(y_int, precision=int_precision), 2), 1)
)
# special cases management
special_case_results = Statement(
# x is sNaN OR y is sNaN
ConditionBlock(
LogicalOr(Test(vx, specifier=Test.IsSignalingNaN), Test(vy, specifier=Test.IsSignalingNaN)),
Return(FP_QNaN(self.precision))
),
# pow(x, ±0) is 1 if x is not a signaling NaN
ConditionBlock(
Test(vy, specifier=Test.IsZero),
Return(Constant(1.0, precision=self.precision))
),
# pow(±0, y) is ±∞ and signals the divideByZero exception for y an odd integer <0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(y_is_odd, vy < 0)),
Return(Select(Test(vx, specifier=Test.IsPositiveZero), FP_PlusInfty(self.precision), FP_MinusInfty(self.precision))),
),
# pow(±0, −∞) is +∞ with no exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_MinusInfty(self.precision)),
),
# pow(±0, +∞) is +0 with no exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is ±0 for finite y>0 an odd integer
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(y_is_odd, vy > 0)),
Return(vx),
),
# pow(−1, ±∞) is 1 with no exception
ConditionBlock(
LogicalAnd(Equal(vx, -1), Test(vy, specifier=Test.IsInfty)),
Return(Constant(1.0, precision=self.precision)),
),
# pow(+1, y) is 1 for any y (even a quiet NaN)
ConditionBlock(
vx == 1,
Return(Constant(1.0, precision=self.precision)),
),
# pow(x, +∞) is +0 for −1<x<1
ConditionBlock(
LogicalAnd(Abs(vx) < 1, Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusZero(self.precision))
),
# pow(x, +∞) is +∞ for x<−1 or for 1<x (including ±∞)
ConditionBlock(
LogicalAnd(Abs(vx) > 1, Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusInfty(self.precision))
),
# pow(x, −∞) is +∞ for −1<x<1
ConditionBlock(
LogicalAnd(Abs(vx) < 1, Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_PlusInfty(self.precision))
),
# pow(x, −∞) is +0 for x<−1 or for 1<x (including ±∞)
ConditionBlock(
LogicalAnd(Abs(vx) > 1, Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_PlusZero(self.precision))
),
# pow(+∞, y) is +0 for a number y < 0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsPositiveInfty), vy < 0),
Return(FP_PlusZero(self.precision))
),
# pow(+∞, y) is +∞ for a number y > 0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsPositiveInfty), vy > 0),
Return(FP_PlusInfty(self.precision))
),
# pow(−∞, y) is −0 for finite y < 0 an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(y_is_odd, vy < 0)),
Return(FP_MinusZero(self.precision)),
),
# pow(−∞, y) is −∞ for finite y > 0 an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(y_is_odd, vy > 0)),
Return(FP_MinusInfty(self.precision)),
),
# pow(−∞, y) is +0 for finite y < 0 and not an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(LogicalNot(y_is_odd), vy < 0)),
Return(FP_PlusZero(self.precision)),
),
# pow(−∞, y) is +∞ for finite y > 0 and not an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(LogicalNot(y_is_odd), vy > 0)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is +∞ and signals the divideByZero exception for finite y<0 and not an odd integer
# TODO: signal divideByZero exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(LogicalNot(y_is_odd), vy < 0)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is +0 for finite y>0 and not an odd integer
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(LogicalNot(y_is_odd), vy > 0)),
Return(FP_PlusZero(self.precision)),
),
)
# manage n=1 separately to avoid catastrophic propagation of errors
# between log2 and exp2 to eventually compute the identity function
# test-case #3
result = Statement(
special_case_results,
# fallback default cases
Return(result_sign * exp2_t1 * exp2_t0_int * exp2_t0_frac))
return result
def numeric_emulate(self, vx, vy):
""" Numeric emulation of pow """
if is_snan(vx) or is_snan(vy):
return FP_QNaN(self.precision)
# pow(x, ±0) is 1 if x is not a signaling NaN
if is_zero(vy):
return 1.0
# pow(±0, y) is ±∞ and signals the divideByZero exception for y an odd integer <0
if is_plus_zero(vx) and not is_special_value(vy) and (int(vy) == vy) and (int(vy) % 2 == 1) and vy < 0:
return FP_PlusInfty(self.precision)
if is_minus_zero(vx) and not is_special_value(vy) and (int(vy) == vy) and (int(vy) % 2 == 1) and vy < 0:
return FP_MinusInfty(self.precision)
# pow(±0, −∞) is +∞ with no exception
if is_zero(vx) and is_minus_zero(vy):
return FP_MinusInfty(self.precision)
# pow(±0, +∞) is +0 with no exception
if is_zero(vx) and is_plus_zero(vy):
return FP_PlusZero(self.precision)
# pow(±0, y) is ±0 for finite y>0 an odd integer
if is_zero(vx) and not is_special_value(vy) and (int(vy) == vy) and (int(vy) % 2 == 1) and vy > 0:
return vx
# pow(−1, ±∞) is 1 with no exception
if vx == -1.0 and is_infty(vy):
return 1
# pow(+1, y) is 1 for any y (even a quiet NaN)
if vx == 1.0:
return 1.0
# pow(x, +∞) is +0 for −1<x<1
if -1 < vx < 1 and is_plus_infty(vy):
return FP_PlusZero(self.precision)
# pow(x, +∞) is +∞ for x<−1 or for 1<x (including ±∞)
if (vx < -1 or vx > 1) and is_plus_infty(vy):
return FP_PlusInfty(self.precision)
# pow(x, −∞) is +∞ for −1<x<1
if -1 < vx < 1 and is_minus_infty(vy):
return FP_PlusInfty(self.precision)
# pow(x, −∞) is +0 for x<−1 or for 1<x (including ±∞)
if is_minus_infty(vy) and (vx < -1 or vx > 1):
return FP_PlusZero(self.precision)
# pow(+∞, y) is +0 for a number y < 0
if is_plus_infty(vx) and vy < 0:
return FP_PlusZero(self.precision)
# pow(+∞, y) is +∞ for a number y > 0
if is_plus_infty(vx) and vy > 0:
return FP_PlusInfty(self.precision)
# pow(−∞, y) is −0 for finite y < 0 an odd integer
if is_minus_infty(vx) and vy < 0 and not is_special_value(vy) and int(vy) == vy and int(vy) % 2 == 1:
return FP_MinusZero(self.precision)
# pow(−∞, y) is −∞ for finite y > 0 an odd integer
if is_minus_infty(vx) and vy > 0 and not is_special_value(vy) and int(vy) == vy and int(vy) % 2 == 1:
return FP_MinusInfty(self.precision)
# pow(−∞, y) is +0 for finite y < 0 and not an odd integer
if is_minus_infty(vx) and vy < 0 and not is_special_value(vy) and not(int(vy) == vy and int(vy) % 2 == 1):
return FP_PlusZero(self.precision)
# pow(−∞, y) is +∞ for finite y > 0 and not an odd integer
if is_minus_infty(vx) and vy > 0 and not is_special_value(vy) and not(int(vy) == vy and int(vy) % 2 == 1):
return FP_PlusInfty(self.precision)
# pow(±0, y) is +∞ and signals the divideByZero exception for finite y<0 and not an odd integer
if is_zero(vx) and vy < 0 and not is_special_value(vy) and not(int(vy) == vy and int(vy) % 2 == 1):
return FP_PlusInfty(self.precision)
# pow(±0, y) is +0 for finite y>0 and not an odd integer
if is_zero(vx) and vy > 0 and not is_special_value(vy) and not(int(vy) == vy and int(vy) % 2 == 1):
return FP_PlusZero(self.precision)
# TODO
# pow(x, y) signals the invalid operation exception for finite x<0 and finite non-integer y.
return sollya.SollyaObject(vx)**sollya.SollyaObject(vy)
def generate_scheme(self):
# We wish to compute vx / vy
vx = self.implementation.add_input_variable(
"x", self.precision, interval=self.input_intervals[0])
vy = self.implementation.add_input_variable(
"y", self.precision, interval=self.input_intervals[1])
# maximum exponent magnitude (to avoid overflow/ underflow during
# intermediary computations
int_prec = self.precision.get_integer_format()
max_exp_mag = Constant(self.precision.get_emax() - 1,
precision=int_prec)
exact_ex = ExponentExtraction(vx,
tag="exact_ex",
precision=int_prec,
debug=debug_multi)
exact_ey = ExponentExtraction(vy,
tag="exact_ey",
precision=int_prec,
debug=debug_multi)
ex = Max(Min(exact_ex, max_exp_mag, precision=int_prec),
-max_exp_mag,
tag="ex",
precision=int_prec)
ey = Max(Min(exact_ey, max_exp_mag, precision=int_prec),
-max_exp_mag,
tag="ey",
precision=int_prec)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Attributes.set_default_silent(True)
# computing the inverse square root
init_approx = None
scaling_factor_x = ExponentInsertion(-ex,
tag="sfx_ei",
precision=self.precision,
debug=debug_multi)
scaling_factor_y = ExponentInsertion(-ey,
tag="sfy_ei",
precision=self.precision,
debug=debug_multi)
def test_interval_out_of_bound_risk(x_range, y_range):
""" Try to determine from x and y's interval if there is a risk
of underflow or overflow """
div_range = abs(x_range / y_range)
underflow_risk = sollya.inf(div_range) < S2**(
self.precision.get_emin_normal() + 2)
overflow_risk = sollya.sup(div_range) > S2**(
self.precision.get_emax() - 2)
return underflow_risk or overflow_risk
out_of_bound_risk = (self.input_intervals[0] is None
or self.input_intervals[1] is None
) or test_interval_out_of_bound_risk(
self.input_intervals[0],
self.input_intervals[1])
Log.report(Log.Debug,
"out_of_bound_risk: {}".format(out_of_bound_risk))
# scaled version of vx and vy, to avoid overflow and underflow
if out_of_bound_risk:
scaled_vx = vx * scaling_factor_x
scaled_vy = vy * scaling_factor_y
scaled_interval = MetaIntervalList(
[MetaInterval(Interval(-2, -1)),
MetaInterval(Interval(1, 2))])
scaled_vx.set_attributes(tag="scaled_vx",
debug=debug_multi,
interval=scaled_interval)
scaled_vy.set_attributes(tag="scaled_vy",
debug=debug_multi,
interval=scaled_interval)
seed_interval = 1 / scaled_interval
print("seed_interval=1/{}={}".format(scaled_interval,
seed_interval))
else:
scaled_vx = vx
scaled_vy = vy
seed_interval = 1 / scaled_vy.get_interval()
# We need a first approximation to 1 / scaled_vy
dummy_seed = ReciprocalSeed(EmptyOperand(precision=self.precision),
precision=self.precision)
if self.processor.is_supported_operation(dummy_seed, self.language):
init_approx = ReciprocalSeed(scaled_vy,
precision=self.precision,
tag="init_approx",
debug=debug_multi)
else:
# generate tabulated version of seed
raise NotImplementedError
current_approx_std = init_approx
# correctly-rounded inverse computation
num_iteration = self.num_iter
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
# check if inputs are zeros
x_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
precision=ML_Bool)
y_zero = Test(vy,
specifier=Test.IsZero,
likely=False,
precision=ML_Bool)
comp_sign = Test(vx,
vy,
specifier=Test.CompSign,
tag="comp_sign",
debug=debug_multi)
# check if divisor is NaN
y_nan = Test(vy, specifier=Test.IsNaN, likely=False, precision=ML_Bool)
# check if inputs are signaling NaNs
x_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
precision=ML_Bool)
y_snan = Test(vy,
specifier=Test.IsSignalingNaN,
likely=False,
precision=ML_Bool)
# check if inputs are infinities
x_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
tag="x_inf",
precision=ML_Bool)
y_inf = Test(vy,
specifier=Test.IsInfty,
likely=False,
tag="y_inf",
debug=debug_multi,
precision=ML_Bool)
scheme = None
gappa_vx, gappa_vy = None, None
# initial reciprocal approximation of 1.0 / scaled_vy
inv_iteration_list, recp_approx = compute_reduced_reciprocal(
init_approx, scaled_vy, self.num_iter)
recp_approx.set_attributes(tag="recp_approx", debug=debug_multi)
# approximation of scaled_vx / scaled_vy
yerr_last, reduced_div_approx, div_iteration_list = compute_reduced_division(
scaled_vx, scaled_vy, recp_approx)
eval_error_range, div_eval_error_range = self.solve_eval_error(
init_approx, recp_approx, reduced_div_approx, scaled_vx, scaled_vy,
inv_iteration_list, div_iteration_list, S2**-7, seed_interval)
eval_error = sup(abs(eval_error_range))
recp_interval = 1 / scaled_vy.get_interval() + eval_error_range
recp_approx.set_interval(recp_interval)
div_interval = scaled_vx.get_interval() / scaled_vy.get_interval(
) + div_eval_error_range
reduced_div_approx.set_interval(div_interval)
reduced_div_approx.set_tag("reduced_div_approx")
if out_of_bound_risk:
unscaled_result = scaling_div_result(reduced_div_approx, ex,
scaling_factor_y,
self.precision)
subnormal_result = subnormalize_result(recp_approx,
reduced_div_approx, ex, ey,
yerr_last, self.precision)
else:
unscaled_result = reduced_div_approx
subnormal_result = reduced_div_approx
x_inf_or_nan = Test(vx, specifier=Test.IsInfOrNaN, likely=False)
y_inf_or_nan = Test(vy,
specifier=Test.IsInfOrNaN,
likely=False,
tag="y_inf_or_nan",
debug=debug_multi)
# generate IEEE exception raising only of libm-compliant
# mode is enabled
enable_raise = self.libm_compliant
# managing special cases
# x inf and y inf
pre_scheme = ConditionBlock(
x_inf_or_nan,
ConditionBlock(
x_inf,
ConditionBlock(
y_inf_or_nan,
Statement(
# signaling NaNs raise invalid operation flags
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)),
),
ConditionBlock(comp_sign,
Return(FP_MinusInfty(self.precision)),
Return(FP_PlusInfty(self.precision)))),
Statement(
ConditionBlock(x_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
x_zero,
ConditionBlock(
LogicalOr(y_zero, y_nan, precision=ML_Bool),
Statement(
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision))), Return(vx)),
ConditionBlock(
y_inf_or_nan,
ConditionBlock(
y_inf,
Return(
Select(comp_sign, FP_MinusZero(self.precision),
FP_PlusZero(self.precision))),
Statement(
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
y_zero,
Statement(
Raise(ML_FPE_DivideByZero)
if enable_raise else Statement(),
ConditionBlock(
comp_sign,
Return(FP_MinusInfty(self.precision)),
Return(FP_PlusInfty(self.precision)))),
# managing numerical value result cases
Statement(
recp_approx,
reduced_div_approx,
ConditionBlock(
Test(unscaled_result,
specifier=Test.IsSubnormal,
likely=False),
# result is subnormal
Statement(
# inexact flag should have been raised when computing yerr_last
# ConditionBlock(
# Comparison(
# yerr_last, 0,
# specifier=Comparison.NotEqual, likely=True),
# Statement(Raise(ML_FPE_Inexact, ML_FPE_Underflow))
#),
Return(subnormal_result), ),
# result is normal
Statement(
# inexact flag should have been raised when computing yerr_last
#ConditionBlock(
# Comparison(
# yerr_last, 0,
# specifier=Comparison.NotEqual, likely=True),
# Raise(ML_FPE_Inexact)
#),
Return(unscaled_result))),
)))))
# managing rounding mode save and restore
# to ensure intermediary computations are performed in round-to-nearest
# clearing exception before final computation
#rnd_mode = GetRndMode()
#scheme = Statement(
# rnd_mode,
# SetRndMode(ML_RoundToNearest),
# yerr_last,
# SetRndMode(rnd_mode),
# unscaled_result,
# ClearException(),
# pre_scheme
#)
scheme = pre_scheme
return scheme
def generate_scalar_scheme(self, vx, n):
# fixing inputs' node tag
vx.set_attributes(tag="x")
n.set_attributes(tag="n")
int_precision = self.precision.get_integer_format()
# assuming x = m.2^e (m in [1, 2[)
# n, positive or null integers
#
# rootn(x, n) = x^(1/n)
# = exp(1/n * log(x))
# = 2^(1/n * log2(x))
# = 2^(1/n * (log2(m) + e))
#
# approximation log2(m)
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
use_reciprocal = False
# non-scaled vx used to compute vx^1
unmodified_vx = vx
is_subnormal = Test(vx, specifier=Test.IsSubnormal, tag="is_subnormal")
exp_correction_factor = self.precision.get_mantissa_size()
mantissa_factor = Constant(2**exp_correction_factor,
tag="mantissa_factor")
vx = Select(is_subnormal, vx * mantissa_factor, vx, tag="corrected_vx")
m = MantissaExtraction(vx, tag="m", precision=self.precision)
e = ExponentExtraction(vx, tag="e", precision=int_precision)
e = Select(is_subnormal,
e - exp_correction_factor,
e,
tag="corrected_e")
ml_log_args = ML_GenericLog.get_default_args(precision=self.precision,
basis=2)
ml_log = ML_GenericLog(ml_log_args)
log_table, log_table_tho, table_index_range = ml_log.generate_log_table(
log_f, inv_approx_table)
log_approx = ml_log.generate_reduced_log_split(
Abs(m, precision=self.precision), log_f, inv_approx_table,
log_table)
# floating-point version of n
n_f = Conversion(n, precision=self.precision, tag="n_f")
inv_n = Division(Constant(1, precision=self.precision), n_f)
log_approx = Select(Equal(vx, 0), FP_MinusInfty(self.precision),
log_approx)
log_approx.set_attributes(tag="log_approx", debug=debug_multi)
if use_reciprocal:
r = Multiplication(log_approx, inv_n, tag="r", debug=debug_multi)
else:
r = Division(log_approx, n_f, tag="r", debug=debug_multi)
# e_n ~ e / n
e_f = Conversion(e, precision=self.precision, tag="e_f")
if use_reciprocal:
e_n = Multiplication(e_f, inv_n, tag="e_n")
else:
e_n = Division(e_f, n_f, tag="e_n")
error_e_n = FMA(e_n, -n_f, e_f, tag="error_e_n")
e_n_int = NearestInteger(e_n, precision=self.precision, tag="e_n_int")
pre_e_n_frac = e_n - e_n_int
pre_e_n_frac.set_attributes(tag="pre_e_n_frac")
e_n_frac = pre_e_n_frac + error_e_n * inv_n
e_n_frac.set_attributes(tag="e_n_frac")
ml_exp2_args = ML_Exp2.get_default_args(precision=self.precision)
ml_exp2 = ML_Exp2(ml_exp2_args)
exp2_r = ml_exp2.generate_scalar_scheme(r, inline_select=True)
exp2_r.set_attributes(tag="exp2_r", debug=debug_multi)
exp2_e_n_frac = ml_exp2.generate_scalar_scheme(e_n_frac,
inline_select=True)
exp2_e_n_frac.set_attributes(tag="exp2_e_n_frac", debug=debug_multi)
exp2_e_n_int = ExponentInsertion(Conversion(e_n_int,
precision=int_precision),
precision=self.precision,
tag="exp2_e_n_int")
n_is_even = Equal(Modulo(n, 2), 0, tag="n_is_even", debug=debug_multi)
n_is_odd = LogicalNot(n_is_even, tag="n_is_odd")
result_sign = Select(
n_is_odd, CopySign(vx, Constant(1.0, precision=self.precision)), 1)
# managing n == -1
if self.expand_div:
ml_division_args = ML_Division.get_default_args(
precision=self.precision, input_formats=[self.precision] * 2)
ml_division = ML_Division(ml_division_args)
self.division_implementation = ml_division.implementation
self.division_implementation.set_scheme(
ml_division.generate_scheme())
ml_division_fct = self.division_implementation.get_function_object(
)
else:
ml_division_fct = Division
# manage n=1 separately to avoid catastrophic propagation of errors
# between log2 and exp2 to eventually compute the identity function
# test-case #3
result = ConditionBlock(
LogicalOr(LogicalOr(Test(vx, specifier=Test.IsNaN), Equal(n, 0)),
LogicalAnd(n_is_even, vx < 0)),
Return(FP_QNaN(self.precision)),
Statement(
ConditionBlock(
Equal(n, -1, tag="n_is_mone"),
#Return(Division(Constant(1, precision=self.precision), unmodified_vx, tag="div_res", precision=self.precision)),
Return(
ml_division_fct(Constant(1, precision=self.precision),
unmodified_vx,
tag="div_res",
precision=self.precision)),
),
ConditionBlock(
# rootn( ±inf, n) is +∞ for even n< 0.
Test(vx, specifier=Test.IsInfty),
Statement(
ConditionBlock(
n < 0,
#LogicalAnd(n_is_odd, n < 0),
Return(
Select(Test(vx,
specifier=Test.IsPositiveInfty),
Constant(FP_PlusZero(self.precision),
precision=self.precision),
Constant(FP_MinusZero(self.precision),
precision=self.precision),
precision=self.precision)),
Return(vx),
), ),
),
ConditionBlock(
# rootn(±0, n) is ±∞ for odd n < 0.
LogicalAnd(LogicalAnd(n_is_odd, n < 0),
Equal(vx, 0),
tag="n_is_odd_and_neg"),
Return(
Select(Test(vx, specifier=Test.IsPositiveZero),
Constant(FP_PlusInfty(self.precision),
precision=self.precision),
Constant(FP_MinusInfty(self.precision),
precision=self.precision),
precision=self.precision)),
),
ConditionBlock(
# rootn( ±0, n) is +∞ for even n< 0.
LogicalAnd(LogicalAnd(n_is_even, n < 0), Equal(vx, 0)),
Return(FP_PlusInfty(self.precision))),
ConditionBlock(
# rootn(±0, n) is +0 for even n > 0.
LogicalAnd(n_is_even, Equal(vx, 0)),
Return(vx)),
ConditionBlock(
Equal(n, 1), Return(unmodified_vx),
Return(result_sign * exp2_r * exp2_e_n_int *
exp2_e_n_frac))))
return result
def standard_test_cases(self):
general_list = [
# ERROR: rootn: inf ulp error at {inf, -2}: *0x0p+0 vs. inf (0x7f800000) at index: 1226
(FP_PlusInfty(self.precision), -2, FP_PlusZero(self.precision)),
# ERROR: rootn: inf ulp error at {inf, -2147483648}: *0x0.0000000000000p+0 vs. inf
(FP_PlusInfty(self.precision), -2147483648,
FP_PlusZero(self.precision)),
#
(FP_PlusZero(self.precision), -1, FP_PlusInfty(self.precision)),
(FP_MinusInfty(self.precision), 1, FP_MinusInfty(self.precision)),
(FP_MinusInfty(self.precision), -1, FP_MinusZero(self.precision)),
# ERROR coucou7: rootn: -inf ulp error at {inf 7f800000, 479638026}: *inf vs. 0x1.000018p+0 (0x3f80000c) at index: 2367
(FP_PlusInfty(self.precision), 479638026,
FP_PlusInfty(self.precision)),
(FP_MinusInfty(self.precision), 479638026),
#(FP_MinusInfty(self.precision), -479638026),
#(FP_PlusInfty(self.precision), -479638026),
# rootn( ±0, n) is ±∞ for odd n< 0.
(FP_PlusZero(self.precision), -1337, FP_PlusInfty(self.precision)),
(FP_MinusZero(self.precision), -1337,
FP_MinusInfty(self.precision)),
# rootn( ±0, n) is +∞ for even n< 0.
(FP_PlusZero(self.precision), -1330, FP_PlusInfty(self.precision)),
# rootn( ±0, n) is +0 for even n> 0.
(FP_PlusZero(self.precision), random.randrange(0, 2**31, 2),
FP_PlusZero(self.precision)),
(FP_MinusZero(self.precision), random.randrange(0, 2**31, 2),
FP_PlusZero(self.precision)),
# rootn( ±0, n) is ±0 for odd n> 0.
(FP_PlusZero(self.precision), random.randrange(1, 2**31, 2),
FP_PlusZero(self.precision)),
(FP_MinusZero(self.precision), random.randrange(1, 2**31, 2),
FP_MinusZero(self.precision)),
# rootn( x, n) returns a NaN for x< 0 and n is even.
(-random.random(), 2 * random.randrange(1, 2**30),
FP_QNaN(self.precision)),
# rootn( x, 0 ) returns a NaN
(random.random(), 0, FP_QNaN(self.precision)),
# vx=nan
(sollya.parse("-nan"), -1811577079, sollya.parse("nan")),
(sollya.parse("-nan"), 832501219, sollya.parse("nan")),
(sollya.parse("-nan"), -857435762, sollya.parse("nan")),
(sollya.parse("-nan"), -1503049611, sollya.parse("nan")),
(sollya.parse("-nan"), 2105620996, sollya.parse("nan")),
#ERROR: rootn: inf ulp error at {-nan, 832501219}: *-nan vs. -0x1.00000df2bed98p+1
#ERROR: rootn: inf ulp error at {-nan, -857435762}: *-nan vs. 0x1.0000000000000p+1
#ERROR: rootn: inf ulp error at {-nan, -1503049611}: *-nan vs. -0x1.0000000000000p+1
#ERROR: rootn: inf ulp error at {-nan, 2105620996}: *-nan vs. 0x1.00000583c4b7ap+1
(sollya.parse("-0x1.cd150ap-105"), 105297051),
(sollya.parse("0x1.ec3bf8p+71"), -1650769017),
# test-case #12
(0.1, 17),
# test-case #11, fails in OpenCL CTS
(sollya.parse("0x0.000000001d600p-1022"), 14),
# test-case #10, fails test with dar(2**-23)
(sollya.parse("-0x1.20aadp-114"), 17),
# test-case #9
(sollya.parse("0x1.a44d8ep+121"), 7),
# test-case #8
(sollya.parse("-0x1.3ef124p+103"), 3),
# test-case #7
(sollya.parse("-0x1.01047ep-2"), 39),
# test-case #6
(sollya.parse("-0x1.0105bp+67"), 23),
# test-case #5
(sollya.parse("0x1.c1f72p+51"), 6),
# special cases
(sollya.parse("0x0p+0"), 1),
(sollya.parse("0x0p+0"), 0),
# test-case #3, catastrophic error for n=1
(sollya.parse("0x1.fc61a2p-121"), 1.0),
# test-case #4 , k=14 < 0 not supported by bigfloat
# (sollya.parse("0x1.ad067ap-66"), -14),
]
# NOTE: expected value assumed 32-bit precision output
fp_32_only = [
#
(sollya.parse("0x1.80bb0ep+70"), 377778829,
sollya.parse("0x1.000002p+0")),
]
# NOTE: the following test-case are only valid if meta-function supports 64-bit integer
# 2nd_input
fp_64_only = [
(sollya.parse("0x1.fffffffffffffp+1023"), -1,
sollya.parse("0x0.4000000000000p-1022")),
(sollya.parse("-0x1.fffffffffffffp1023"), -1,
sollya.parse("-0x0.4000000000000p-1022")),
#(sollya.parse("-0x1.fffffffffffffp+1023"), 1),
#(sollya.parse("0x1.fffffffffffffp+1023"), -1),
# ERROR coucou8: rootn: inf ulp error at {-inf, 1854324695}: *-inf vs. -0x1.0000066bfdd60p+0
(FP_MinusInfty(self.precision), 1854324695,
FP_MinusInfty(self.precision)),
# ERROR: rootn: -60.962402 ulp error at {0x0.000000001d600p-1022, 14}: *0x1.67d4ff97d1fd9p-76 vs. 0x1.67d4ff97d1f9cp-76
(sollya.parse("0x0.000000001d600p-1022"), 14,
sollya.parse("0x1.67d4ff97d1fd9p-76")),
# ERROR: rootn: -430452000.000000 ulp error at {0x1.ffffffff38c00p-306, 384017876}: *0x1.ffffed870ff01p-1 vs. 0x1.ffffebec8d1d2p-1
(sollya.parse("0x1.ffffffff38c00p-306"), 384017876,
sollya.parse("0x1.ffffed870ff01p-1")), # vs. 0x1.ffffebec8d1d2p-1
# ERROR: rootn: 92996584.000000 ulp error at {0x1.ffffffffdae80p-858, -888750231}: *0x1.00000b36b1173p+0 vs. 0x1.00000b8f6155ep+0
(sollya.parse("0x1.ffffffffdae80p-858"), -888750231,
sollya.parse("0x1.00000b36b1173p+0")),
# ERROR: rootn: 379474.906250 ulp error at {0x0.0000000000022p-1022, -1538297900}: *0x1.00000814a68ffp+0 vs. 0x1.0000081503352p+0
(sollya.parse("0x0.00000006abfffp-1022"), -1221802473,
sollya.parse("0x1.00000a01818a4p+0")),
(sollya.parse("0x1.ffffffffd0a00p-260"), 1108043946,
sollya.parse("0x1.fffffa9042997p-1")),
(sollya.parse("0x1.3fffffffff1c0p-927"), -1997086266,
sollya.parse("0x1.0000056564c5ep+0")),
(sollya.parse("0x1.ffffffff38c00p-306"), 384017876,
sollya.parse("0x1.ffffed870ff01p-1")),
(sollya.parse("0x0.15c000000002ap-1022"), 740015941,
sollya.parse("0x1.ffffdfc47b57ep-1")),
(sollya.parse("0x0.00000000227ffp-1022"), -1859058847,
sollya.parse("0x1.0000069c7a01bp+0")),
(sollya.parse("0x0.0568000000012p-1022"), -447352599,
sollya.parse("0x1.00001ab640c38p+0")),
(sollya.parse("0x0.000000000000dp-1022"), 132283432,
sollya.parse("0x1.ffff43d1db82ap-1")),
(sollya.parse("-0x1.c80000000026ap+1023"), 275148531,
sollya.parse("-0x1.00002b45a7314p+0")),
(sollya.parse("0x0.022200000000ep-1022"), -1969769414,
sollya.parse("0x1.000006130e858p+0")),
(sollya.parse("0x0.0000000000011p-1022"), 851990770,
sollya.parse("0x1.ffffe2cafaff6p-1")),
(sollya.parse("0x1.8fffffffff348p-1010"), 526938360,
sollya.parse("0x1.ffffd372e2b81p-1")),
(sollya.parse("0x0.0000000000317p-1022"), -1315106194,
sollya.parse("0x1.0000096973ac9p+0")),
(sollya.parse("0x1.1ffffffff2d20p-971"), 378658008,
sollya.parse("0x1.ffffc45e803b2p-1")),
#
(sollya.parse("0x0.0568000000012p-1022"), -447352599,
sollya.parse("0x1.00001ab640c38p+0")),
#
(sollya.parse("0x1.ffffffffd0a00p-260"), 1108043946,
sollya.parse("0x1.fffffa9042997p-1")),
(FP_MinusZero(self.precision), -21015979,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -85403731,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -180488973,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -1365227287,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -1802885579,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -1681209663,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -1152797721,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -1614890585,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -812655517,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -628647891,
FP_MinusInfty(self.precision)),
(sollya.parse("0x1.ffffffffdae80p-858"), -888750231,
sollya.parse("0x1.00000b36b1173p+0")),
(sollya.parse("0x0.0568000000012p-1022"), -447352599,
sollya.parse("0x1.00001ab640c38p+0")),
(sollya.parse("0x0.00000006abfffp-1022"), -1221802473,
sollya.parse("0x1.00000a01818a4p+0")),
(sollya.parse("0x0.0000000000022p-1022"), -1538297900,
sollya.parse("0x1.00000814a68ffp+0")),
#ERROR: rootn: inf ulp error at {-0x0.0000000000000p+0, -1889147085}: *-inf vs. inf
#ERROR: rootn: inf ulp error at {-0x0.0000000000000p+0, -373548013}: *-inf vs. inf
(FP_MinusZero(self.precision), -1889147085,
FP_MinusInfty(self.precision)),
(FP_MinusZero(self.precision), -373548013,
FP_MinusInfty(self.precision)),
#ERROR: rootn: inf ulp error at {-0x0.0000000000000p+0, -1889147085}: *-inf vs. inf
#ERROR: rootn: inf ulp error at {-0x0.0000000000000p+0, -373548013}: *-inf vs. inf
# [email protected]: PE 0: error[84]: ml_rootn(-0x1.b1a6765727e72p-902, -7.734955e+08/-773495525), result is -0x1.00000d8cb5b3cp+0 vs expected [nan;nan]
(sollya.parse("-0x1.b1a6765727e72p-902"), -773495525),
# ERROR: rootn: -40564819207303340847894502572032.000000 ulp error at {-0x0.fffffffffffffp-1022, 1}: *-0x0.fffffffffffffp-1022 vs. -0x1.ffffffffffffep-970
(sollya.parse("-0x0.fffffffffffffp-1022 "), 1,
sollya.parse("-0x0.fffffffffffffp-1022 ")),
# ERROR: rootn: 1125899906842624.000000 ulp error at {-0x1.fffffffffffffp+1023, -1}: *-0x0.4000000000000p-1022 vs. -0x0.0000000000000p+0
(sollya.parse("-0x1.fffffffffffffp+1023"), -1,
sollya.parse("-0x0.4000000000000p-1022")),
(sollya.parse("0x1.fffffffffffffp+1023"), -1,
sollya.parse("0x0.4000000000000p-1022")),
]
return (fp_64_only if self.precision.get_bit_size() >= 64 else []) \
+ (fp_32_only if self.precision.get_bit_size() == 32 else []) \
+ general_list
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
vx.set_attributes(precision=self.precision,
tag="vx",
debug=debug_multi)
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 SqrtRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
C0 = Constant(0, precision=self.precision)
C0_plus = Constant(FP_PlusZero(self.precision))
test_NaN = Test(vx,
specifier=Test.IsNaN,
likely=False,
debug=debug_multi,
tag="is_NaN",
precision=ML_Bool)
test_negative = Comparison(vx,
C0,
specifier=Comparison.Less,
debug=debug_multi,
tag="is_Negative",
precision=ML_Bool,
likely=False)
test_zero = Comparison(vx,
C0_plus,
specifier=Comparison.Equal,
likely=False,
debug=debug_multi,
tag="Is_Zero",
precision=ML_Bool)
test_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debug_multi,
tag="is_Inf",
precision=ML_Bool)
test_NaN_or_Neg = LogicalOr(test_NaN,
test_negative,
precision=ML_Bool,
likely=False)
test_NaN_or_Inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="is_nan_or_inf",
precision=ML_Bool)
test_negative_or_zero = Comparison(vx,
C0,
specifier=Comparison.LessOrEqual,
debug=debug_multi,
tag="is_Negative_or_zero",
precision=ML_Bool,
likely=False)
test_std = LogicalNot(LogicalOr(test_NaN_or_Inf,
test_negative_or_zero,
precision=ML_Bool,
likely=False),
precision=ML_Bool,
likely=True)
return_PosZero = Statement(Return(FP_PlusInfty(self.precision)))
return_NegZero = Statement(Return(FP_MinusInfty(self.precision)))
return_NaN_or_neg = Statement(Return(FP_QNaN(self.precision)))
return_inf = Statement(Return(C0))
NR_init = ReciprocalSquareRootSeed(vx,
precision=self.precision,
tag="sqrt_seed",
debug=debug_multi)
result = compute_isqrt(vx, NR_init, self.num_iter, self.precision)
return_non_std = ConditionBlock(
test_NaN_or_Neg, return_NaN_or_neg,
ConditionBlock(
test_inf, return_inf,
ConditionBlock(test_zero, return_PosZero, return_NegZero)))
scheme = Statement(
ConditionBlock(test_std, Statement(Return(result)),
Statement(return_non_std)))
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
sollya_precision = self.get_input_precision().get_sollya_object()
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
# 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)))
log2_hi_value = round(
log_f(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), RN)
log2_lo_value = round(
log_f(2) - log2_hi_value, self.precision.sollya_object, RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
int_precision = self.precision.get_integer_format()
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debug_multi)
#---------------------
# Approximation scheme
#---------------------
# log10(x) = log10(m.2^e) = log10(m.2^(e-t+t))
# = log10(m.2^-t) + (e+t) log10(2)
# t = (m > sqrt(2)) ? 1 : 0 is used to avoid catastrophic cancellation
# when e = -1 and m ~ 2
#
#
# log10(m.2^-t) = log10(m.r/r.2^-t) = log10(m.r) + log10(2^-t/r)
# = log10(m.r) - log10(r.2^t)
# where r = rcp(m) an approximation of 1/m such that r.m ~ 1
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = inv_approx_table.index_size
table_index_range = range(1, 2**table_index_size)
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision)
log_table_tho = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision)
log_table[0][0] = 0.0
log_table[0][1] = 0.0
log_table_tho[0][0] = 0.0
log_table_tho[0][1] = 0.0
hi_size = self.precision.get_field_size() - (
self.precision.get_exponent_size() + 1)
for i in table_index_range:
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
#inv_value = (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
inv_value = inv_approx_table[i]
value_high = round(log_f(inv_value), hi_size, sollya.RN)
value_low = round(
log_f(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
inv_value_tho = S2 * inv_approx_table[i]
value_high_tho = round(log_f(inv_value_tho), hi_size, sollya.RN)
value_low_tho = round(
log_f(inv_value_tho) - value_high_tho, sollya_precision,
sollya.RN)
log_table_tho[i][0] = value_high_tho
log_table_tho[i][1] = value_low_tho
# determining log_table range
high_index_function = lambda table, i: table[i][0]
low_index_function = lambda table, i: table[i][1]
table_high_interval = log_table.get_subset_interval(
high_index_function, table_index_range)
table_low_interval = log_table.get_subset_interval(
low_index_function, table_index_range)
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_multi)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debug_multi)
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debug_multi)
tho_cond = _vx_mant > Constant(sollya.sqrt(2),
precision=self.precision)
tho = Select(tho_cond,
Constant(1.0, precision=self.precision),
Constant(0.0, precision=self.precision),
precision=self.precision,
tag="tho",
debug=debug_multi)
rcp = ReciprocalSeed(_vx_mant, precision=self.precision, tag="rcp")
r = Multiplication(rcp,
_vx_mant,
precision=self.precision,
tag="r")
int_format = self.precision.get_integer_format()
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(ReciprocalSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_multi,
silent=True),
precision=int_format),
Constant(-2, precision=int_format),
precision=int_format),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_multi)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_multi)
_red_vx = arg_red_index * _vx_mant - 1.0
inv_err = S2**-6
red_interval = Interval(1 - inv_err, 1 + inv_err)
_red_vx.set_attributes(tag="_red_vx",
debug=debug_multi,
interval=red_interval)
# return in case of standard (non-special) input
_log_inv_lo = Select(tho_cond,
TableLoad(log_table_tho, table_index, 1),
TableLoad(log_table, table_index, 1),
tag="log_inv_lo",
debug=debug_multi)
_log_inv_hi = Select(tho_cond,
TableLoad(log_table_tho, table_index, 0),
TableLoad(log_table, table_index, 0),
tag="log_inv_hi",
debug=debug_multi)
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + x) / x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
Log.report(Log.Info, "generating polynomial evaluation scheme")
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Info, poly_object.get_sollya_object())
corr_exp = Conversion(_vx_exp if exp_corr_factor == None else
_vx_exp + exp_corr_factor,
precision=self.precision) + tho
corr_exp.set_attributes(tag="corr_exp", debug=debug_multi)
# _poly approximates log10(1+r)/r
# _poly * red_vx approximates log10(x)
m0h, m0l = Mul211(_red_vx, _poly)
m0h, m0l = Add212(_red_vx, m0h, m0l)
m0h.set_attributes(tag="m0h", debug=debug_multi)
m0l.set_attributes(tag="m0l")
l0_h = corr_exp * log2_hi
l0_l = corr_exp * log2_lo
l0_h.set_attributes(tag="l0_h")
l0_l.set_attributes(tag="l0_l")
rh, rl = Add222(l0_h, l0_l, m0h, m0l)
rh.set_attributes(tag="rh0", debug=debug_multi)
rl.set_attributes(tag="rl0", debug=debug_multi)
rh, rl = Add222(-_log_inv_hi, -_log_inv_lo, rh, rl)
rh.set_attributes(tag="rh", debug=debug_multi)
rl.set_attributes(tag="rl", debug=debug_multi)
if sollya.log(self.basis) != 1.0:
lbh = self.precision.round_sollya_object(
1 / sollya.log(self.basis))
lbl = self.precision.round_sollya_object(
1 / sollya.log(self.basis) - lbh)
rh, rl = Mul222(rh, rl, lbh, lbl)
return rh
else:
return rh
result = compute_log(vx)
result.set_attributes(tag="result", debug=debug_multi)
if False:
# building eval error map
eval_error_map = {
red_vx:
Variable("red_vx",
precision=self.precision,
interval=red_vx.get_interval()),
log_inv_hi:
Variable("log_inv_hi",
precision=self.precision,
interval=table_high_interval),
log_inv_lo:
Variable("log_inv_lo",
precision=self.precision,
interval=table_low_interval),
corr_exp:
Variable("corr_exp_g",
precision=self.precision,
interval=self.precision.get_exponent_interval()),
}
# computing gappa error
if is_gappa_installed():
poly_eval_error = self.get_eval_error(result, eval_error_map)
Log.report(Log.Info, "poly_eval_error: ", poly_eval_error)
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debug_multi,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debug_multi,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debug_multi,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debug_multi,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debug_multi,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debug_multi,
likely=False)
# exp=-1 case
Log.report(Log.Info, "managing exp=-1 case")
#red_vx_2 = arg_red_index * vx_mant * 0.5
#approx_interval2 = Interval(0.5 - inv_err, 0.5 + inv_err)
#poly_degree2 = sup(guessdegree(log(x), approx_interval2, S2**-(self.precision.get_field_size()+1))) + 1
#poly_object2 = Polynomial.build_from_approximation(log(sollya.x), poly_degree, [self.precision]*(poly_degree+1), approx_interval2, sollya.absolute)
#print "poly_object2: ", poly_object2.get_sollya_object()
#poly2 = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object2, red_vx_2, unified_precision = self.precision)
#poly2.set_attributes(tag = "poly2", debug = debug_multi)
#result2 = (poly2 - log_inv_hi - log_inv_lo)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal = compute_log(vx * S2100, exp_corr_factor=m100)
# main scheme
Log.report(Log.Info, "MDL scheme")
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
), Return(result_subnormal)), Return(result))))
scheme = pre_scheme
return scheme
def generate_scalar_scheme(self, vx):
# 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)))
int_precision = self.precision.get_integer_format()
vx_exp = ExponentExtraction(vx,
tag="vx_exp",
precision=int_precision,
debug=debug_multi)
#---------------------
# Approximation scheme
#---------------------
# log(x) = log(m.2^e) = log(m.2^(e-tho+tho))
# = log(m.2^-tho) + (e+tho) log(2)
# tho = (m > sqrt(2)) ? 1 : 0 is used to avoid catastrophic cancellation
# when e = -1 and m ~ 2
#
#
# log(m.2^-tho) = log(m.r/r.2^-tho) = log(m.r) + log(2^-tho/r)
# = log(m.r) - log(r.2^tho)
# where r = rcp(m) an approximation of 1/m such that r.m ~ 1
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
# table of the reciprocal approximation of the targeted processor
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
log_table, log_table_tho, table_index_range = self.generate_log_table(
log_f, inv_approx_table)
# determining log_table range
high_index_function = lambda table, i: table[i][0]
low_index_function = lambda table, i: table[i][1]
table_high_interval = log_table.get_subset_interval(
high_index_function, table_index_range)
table_low_interval = log_table.get_subset_interval(
low_index_function, table_index_range)
result = self.generate_reduced_log(vx, log_f, inv_approx_table,
log_table, log_table_tho)
result.set_attributes(tag="result", debug=debug_multi)
if False:
# building eval error map
eval_error_map = {
red_vx:
Variable("red_vx",
precision=self.precision,
interval=red_vx.get_interval()),
log_inv_hi:
Variable("log_inv_hi",
precision=self.precision,
interval=table_high_interval),
log_inv_lo:
Variable("log_inv_lo",
precision=self.precision,
interval=table_low_interval),
corr_exp:
Variable("corr_exp_g",
precision=self.precision,
interval=self.precision.get_exponent_interval()),
}
# computing gappa error
if is_gappa_installed():
poly_eval_error = self.get_eval_error(result, eval_error_map)
Log.report(Log.Info, "poly_eval_error: ", poly_eval_error)
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debug_multi,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debug_multi,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debug_multi,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debug_multi,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debug_multi,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debug_multi,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debug_multi,
likely=False)
# exp=-1 case
Log.report(Log.Info, "managing exp=-1 case")
#red_vx_2 = arg_red_index * vx_mant * 0.5
#approx_interval2 = Interval(0.5 - inv_err, 0.5 + inv_err)
#poly_degree2 = sup(guessdegree(log(x), approx_interval2, S2**-(self.precision.get_field_size()+1))) + 1
#poly_object2 = Polynomial.build_from_approximation(log(sollya.x), poly_degree, [self.precision]*(poly_degree+1), approx_interval2, sollya.absolute)
#print "poly_object2: ", poly_object2.get_sollya_object()
#poly2 = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object2, red_vx_2, unified_precision = self.precision)
#poly2.set_attributes(tag = "poly2", debug = debug_multi)
#result2 = (poly2 - log_inv_hi - log_inv_lo)
m100 = Constant(-100, precision=int_precision)
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal = self.generate_reduced_log(vx * S2100,
log_f,
inv_approx_table,
log_table,
log_table_tho,
exp_corr_factor=m100)
# main scheme
Log.report(Log.Info, "MDL scheme")
pre_scheme = ConditionBlock(
neg_input,
Statement(
ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision), precision=self.precision)),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision),
precision=self.precision),
),
Statement(
ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision),
precision=self.precision))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision),
precision=self.precision),
), Return(result_subnormal)), Return(result))))
scheme = pre_scheme
return scheme
def generate_scheme(self):
#func_implementation = CodeFunction(self.function_name, output_format = self.precision)
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
sollya_precision = self.get_input_precision().get_sollya_object()
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
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)))
log2_hi_value = round(
log10(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), RN)
log2_lo_value = round(
log10(2) - log2_hi_value, self.precision.sollya_object, RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = self.precision.get_integer_format()
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
table_index_range = range(1, 2**table_index_size)
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision)
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in table_index_range:
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
#inv_value = (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
inv_value = inv_approx_table[i][0]
value_high = round(
log10(inv_value),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(
log10(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
# determining log_table range
high_index_function = lambda table, i: table[i][0]
low_index_function = lambda table, i: table[i][1]
table_high_interval = log_table.get_subset_interval(
high_index_function, table_index_range)
table_low_interval = log_table.get_subset_interval(
low_index_function, table_index_range)
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
debug=debug_lftolx)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debugd)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(_vx_mant, precision=int_precision, debug=debuglx),
self.precision.get_field_size() - 7,
debug=debuglx),
0x7f,
tag="table_index",
debug=debuglld)
# argument reduction
# 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)
#if not processor.is_supported_operation(arg_red_index):
# if self.precision != ML_Binary32:
# arg_red_index = DivisionSeed(Conversion(_vx_mant, precision = ML_Binary32), precision = ML_Binary32,
_red_vx = arg_red_index * _vx_mant - 1.0
inv_err = S2**-7
red_interval = Interval(1 - inv_err, 1 + inv_err)
_red_vx.set_attributes(tag="_red_vx",
debug=debug_lftolx,
interval=red_interval)
# 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)
print("building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log10(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log10(1 + x) / x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object #.sub_poly(start_index = 1)
print("generating polynomial evaluation scheme")
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_lftolx)
print(global_poly_object.get_sollya_object())
corr_exp = Conversion(
_vx_exp if exp_corr_factor == None else _vx_exp +
exp_corr_factor,
precision=self.precision)
split_red_vx = Split(_red_vx,
precision=ML_DoubleDouble,
tag="split_red_vx",
debug=debug_ddtolx)
red_vx_hi = split_red_vx.hi
red_vx_lo = split_red_vx.lo
# result = _red_vx * poly - log_inv_hi - log_inv_lo + _vx_exp * log2_hi + _vx_exp * log2_lo
pre_result = -_log_inv_hi + ((_red_vx * _poly +
(corr_exp * log2_lo - _log_inv_lo)))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_hex",
debug=debug_lftolx)
cancel_part = (corr_exp * log2_hi - _log_inv_hi)
cancel_part.set_attributes(tag="cancel_part", debug=debug_lftolx)
sub_part = red_vx_hi + cancel_part
sub_part.set_attributes(tag="sub_part", debug=debug_lftolx)
#result_one_low_part = (red_vx_hi * _poly + (red_vx_lo + (red_vx_lo * _poly + (corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part = ((red_vx_lo +
(red_vx_lo * _poly +
(corr_exp * log2_lo - _log_inv_lo))))
result_one_low_part.set_attributes(tag="result_one_low_part",
debug=debug_lftolx)
_result_one = (
(sub_part) + red_vx_hi * _poly) + result_one_low_part
_result = exact_log2_hi_exp + pre_result
return _result, _poly, _log_inv_lo, _log_inv_hi, _red_vx, _result_one, corr_exp
result, poly, log_inv_lo, log_inv_hi, red_vx, new_result_one, corr_exp = compute_log(
vx)
result.set_attributes(tag="result", debug=debug_lftolx)
new_result_one.set_attributes(tag="new_result_one", debug=debug_lftolx)
# building eval error map
eval_error_map = {
red_vx:
Variable("red_vx",
precision=self.precision,
interval=red_vx.get_interval()),
log_inv_hi:
Variable("log_inv_hi",
precision=self.precision,
interval=table_high_interval),
log_inv_lo:
Variable("log_inv_lo",
precision=self.precision,
interval=table_low_interval),
corr_exp:
Variable("corr_exp_g",
precision=self.precision,
interval=self.precision.get_exponent_interval()),
}
# computing gappa error
if is_gappa_installed():
poly_eval_error = self.get_eval_error(result, eval_error_map)
print("poly_eval_error: ", poly_eval_error)
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debugd,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debugd,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debugd)
# exp=-1 case
print("managing exp=-1 case")
#red_vx_2 = arg_red_index * vx_mant * 0.5
#approx_interval2 = Interval(0.5 - inv_err, 0.5 + inv_err)
#poly_degree2 = sup(guessdegree(log(x), approx_interval2, S2**-(self.precision.get_field_size()+1))) + 1
#poly_object2 = Polynomial.build_from_approximation(log(sollya.x), poly_degree, [self.precision]*(poly_degree+1), approx_interval2, sollya.absolute)
#print "poly_object2: ", poly_object2.get_sollya_object()
#poly2 = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object2, red_vx_2, unified_precision = self.precision)
#poly2.set_attributes(tag = "poly2", debug = debug_lftolx)
#result2 = (poly2 - log_inv_hi - log_inv_lo)
log_subtract = -log_inv_hi - log2_hi
log_subtract.set_attributes(tag="log_subtract", debug=debug_lftolx)
result2 = (log_subtract) + ((poly * red_vx) - (log_inv_lo + log2_lo))
result2.set_attributes(tag="result2", debug=debug_lftolx)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
print("managing close to 1.0 cases")
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log10(1 + sollya.x) / sollya.x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log10(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
sollya.absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_lftolx)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one", debug=debugd, likely=False)
# main scheme
print("MDL scheme")
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
), Return(result_subnormal)),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result2),
Return(result))
#ConditionBlock(cond_one,
#Return(new_result_one),
#ConditionBlock(exp_mone,
#Return(result2),
#Return(result)
#)
#)
))))
scheme = pre_scheme
return scheme