def generate_scheme(self):
#func_implementation = CodeFunction(self.function_name, output_format = self.precision)
vx = self.implementation.add_input_variable("x", self.precision)
table_size = 16
row_size = 2
new_table = ML_NewTable(dimensions=[table_size, row_size],
storage_precision=self.precision)
for i in range(table_size):
new_table[i][0] = i
new_table[i][1] = i + 1
index = Modulo(vx,
Constant(table_size, precision=ML_Int32),
precision=ML_Int32)
load_value_lo = TableLoad(new_table,
index,
Constant(0, precision=ML_Int32),
precision=self.precision)
load_value_hi = TableLoad(new_table,
index,
Constant(1, precision=ML_Int32),
precision=self.precision)
Log.report(Log.Info,
"table interval: {}".format(new_table.get_interval()))
out_table = ML_NewTable(dimensions=[table_size],
storage_precision=self.precision,
empty=True)
result = Addition(load_value_lo,
load_value_hi,
precision=self.precision)
scheme = Statement(
TableStore(
result,
out_table,
Constant(13, precision=ML_Int32),
precision=ML_Void,
),
Return(
TableLoad(out_table,
Constant(13, precision=ML_Int32),
precision=self.precision),
precision=self.precision,
))
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", ML_Binary32)
px = self.implementation.add_input_variable("px", ML_Binary32_p)
result = vx * vx
# pointer dereferencing and value assignment
px_assign = ReferenceAssign(Dereference(px, precision=ML_Binary32),
result)
# pointer to pointer cast
py = Variable("py", precision=ML_Binary64_p, vartype=Variable.Local)
py_assign = ReferenceAssign(py, TypeCast(px, precision=ML_Binary64_p))
table_size = 16
row_size = 2
new_table = ML_NewTable(dimensions=[table_size, row_size],
storage_precision=self.precision)
for i in range(table_size):
new_table[i][0] = i
new_table[i][1] = i + 1
# cast between table and pointer
pz = Variable("pz",
precision=ML_Pointer_Format(self.precision),
vartype=Variable.Local)
pz_assign = ReferenceAssign(
pz, TypeCast(new_table, precision=ML_Binary64_p))
scheme = Statement(px_assign, py_assign, pz_assign)
return scheme
def generate_scheme(self):
# declaring function input variable
vx = self.implementation.add_input_variable("x", self.precision)
add_xx = Addition(vx, vx, precision=self.precision)
mult = Multiplication(add_xx, vx, precision=self.precision)
cst = Constant(1.1, precision=self.precision)
index_size = 4
table_size = 2**index_size
table = ML_NewTable(dimensions=[table_size],
storage_precision=self.precision)
for i in range(table_size):
table[i] = i
index = NearestInteger(vx, precision=ML_Int32)
# index = index % table_size = index & (2**index_size - 1)
index = BitLogicAnd(index,
Constant(2**index_size - 1, precision=ML_Int32),
precision=ML_Int32)
index = BitLogicRightShift(index,
Constant(1, precision=ML_Int32),
precision=ML_Int32)
table_value = TableLoad(table, index, precision=self.precision)
int_tree = Multiplication(index,
Addition(index,
Constant(7, precision=ML_Int32),
precision=ML_Int32),
precision=ML_Int32)
result = Multiplication(
table_value,
FusedMultiplyAdd(Addition(cst,
Conversion(int_tree,
precision=self.precision),
precision=self.precision),
mult,
add_xx,
specifier=FusedMultiplyAdd.Subtract,
precision=self.precision),
precision=self.precision,
tag="result")
scheme = Return(result, precision=self.precision, debug=debug_multi)
# conv_pass = Pass_M128_Promotion(self.processor)
# new_scheme = conv_pass.execute(scheme)
return scheme
def generate_log_table(self, log_f, inv_approx_table):
""" generate 2 tables:
log_table[i] = 2-word unevaluated sum approximation of log_f(inv_approx_table[i])
log_table_tho[i] = 2-word unevaluated sum approximation of log_f(2*inv_approx_table[i])
"""
sollya_precision = self.get_input_precision().get_sollya_object()
# 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,
const=True)
log_table_tho = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
const=True)
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 = 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
return log_table, log_table_tho, table_index_range
def generate_scheme(self):
# declaring function input variable
vx = self.implementation.add_input_variable("x", self.get_input_precision(0))
bf16_params = ML_NewTable(dimensions=[self.table_size], storage_precision=BFloat16)
for i in range(self.table_size):
bf16_params[i] = 1.1**i
conv_vx = Conversion(TableLoad(bf16_params, vx), precision=ML_Binary32, tag="conv_vx", debug=debug_multi)
result = conv_vx
scheme = Return(result, precision=self.precision, debug=debug_multi)
return scheme
def generate_1d_table(dim,
storage_precision,
tag,
value_gen=lambda index: None,
empty=False,
const=True):
""" generate a 1D ML_NewTable by using the given value generator @p value_gen """
gen_table = ML_NewTable(dimensions=[dim],
storage_precision=storage_precision,
tag=tag,
const=const,
empty=empty)
for i in range(dim):
gen_table[i] = value_gen(i)
return gen_table
def generate_2d_table(dim0,
dim1,
storage_precision,
tag,
value_gen=(lambda index0: None),
const=True):
""" generate a 2D ML_NewTable by using the given value generator @p value_gen,
values are generated one row at a time (rather than cell by cell) """
gen_table = ML_NewTable(dimensions=[dim0, dim1],
storage_precision=storage_precision,
const=const,
tag=tag)
for i0 in range(dim0):
row_values = value_gen(i0)
for i1 in range(dim1):
gen_table[i0][i1] = row_values[i1]
return gen_table
def generate_2d_multi_table(size_offset_list,
dim1,
storage_precision,
tag,
value_gen=lambda table_index, sub_row_index: None):
""" generate a 2D multi-array stored in a ML_NewTable.
The multi-array dimensions are defined by the (size, offset) pairs in size_offset_list
for the first dimension and @p dim1 for the second dimension.
Table value are obtained by using the given value generator @p value_gen,
values are generated one row at a time (rather than cell by cell) """
# table first dimension is the sum of each sub-array size
dim0 = sum(size_offset_list[sub_id][0]
for sub_id in range(size_offset_list.dimensions[0]))
gen_table = ML_NewTable(dimensions=[dim0, dim1],
storage_precision=storage_precision,
tag=tag)
for table_index, (size, offset) in enumerate(size_offset_list):
for i0 in range(size):
row_values = value_gen(table_index, i0)
for i1 in range(dim1):
gen_table[offset + i0][i1] = row_values[i1]
return gen_table
def generate_scalar_scheme(self, vx, inline_select=False):
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# r_interval = Interval(0, 1.0)
index_size = 3
r_interval = Interval(-2**(-index_size), 2**-index_size)
local_ulp = sup(ulp(2**r_interval, self.precision))
Log.report(Log.Info, "ulp: ", local_ulp)
error_goal = S2**-1 * local_ulp
Log.report(Log.Info, "error goal: ", error_goal)
sollya_precision = {
ML_Binary32: sollya.binary32,
ML_Binary64: sollya.binary64
}[self.precision]
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
# Argument Reduction
# r = x - floor(x), r >= 0
vx_floor = Floor(vx,
precision=self.precision,
tag='vx_floor',
debug=debug_multi)
vx_int = Conversion(vx_floor,
precision=int_precision,
tag="vx_int",
debug=debug_multi)
vx_intf = vx_floor # Conversion(vx_int, precision = self.precision)
vx_r = vx - vx_intf
r_hi = NearestInteger(vx_r * 2**index_size,
precision=self.precision,
tag="r_hi",
debug=debug_multi)
# clamping r_hi_int within table-size to make sure
# it does not exceeds hi_part_table when used to index it
r_hi_int = Max(
Min(
Conversion(r_hi,
precision=int_precision,
tag="r_hi_int",
debug=debug_multi), 2**index_size + 1), 0)
r_lo = vx_r - r_hi * 2**-index_size
r_lo.set_attributes(tag="r_lo", debug=debug_multi)
vx_r.set_attributes(tag="vx_r", debug=debug_multi)
degree = sup(guessdegree(2**(sollya.x), r_interval, error_goal)) + 2
precision_list = [1] + [self.precision] * degree
exp_X = ExponentInsertion(vx_int,
tag="exp_X",
debug=debug_multi,
precision=self.precision)
#Polynomial Approx
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_object, poly_error = Polynomial.build_from_approximation_with_error(
2**(sollya.x) - 1, degree, precision_list, r_interval,
sollya.absolute)
Log.report(Log.Info, "Poly : %s" % poly_object)
Log.report(Log.Info, "poly_error : ", poly_error)
poly = polynomial_scheme_builder(poly_object.sub_poly(start_index=1),
r_lo,
unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
hi_part_table = ML_NewTable(dimensions=[2**index_size + 1],
storage_precision=self.precision,
tag=self.uniquify_name("exp2_table"),
const=True)
for i in range(2**index_size + 1):
input_value = i * 2**-index_size
tab_value = self.precision.round_sollya_object(
sollya.SollyaObject(2)**(input_value))
hi_part_table[i] = tab_value
hi_part_value = TableLoad(hi_part_table,
r_hi_int,
precision=self.precision,
tag="hi_part_value",
debug=debug_multi)
#Handling special cases
oflow_bound = Constant(self.precision.get_emax() + 1,
precision=self.precision)
subnormal_bound = self.precision.get_emin_subnormal()
uflow_bound = self.precision.get_emin_normal()
Log.report(Log.Info, "oflow : ", oflow_bound)
#print "uflow : ", uflow_bound
#print "sub : ", subnormal_bound
test_overflow = Comparison(vx,
oflow_bound,
specifier=Comparison.GreaterOrEqual)
test_overflow.set_attributes(tag="oflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_underflow = Comparison(vx, uflow_bound, specifier=Comparison.Less)
test_underflow.set_attributes(tag="uflow_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
test_subnormal = Comparison(vx,
subnormal_bound,
specifier=Comparison.Greater)
test_subnormal.set_attributes(tag="sub_test",
debug=debug_multi,
likely=False,
precision=ML_Bool)
subnormal_offset = -(uflow_bound - vx_int)
subnormal_offset.set_attributes(tag="offset", debug=debug_multi)
exp_offset = ExponentInsertion(subnormal_offset,
precision=self.precision,
debug=debug_multi,
tag="exp_offset")
exp_min = ExponentInsertion(uflow_bound,
precision=self.precision,
debug=debug_multi,
tag="exp_min")
subnormal_result = hi_part_value * exp_offset * exp_min * poly + hi_part_value * exp_offset * exp_min
test_std = LogicalOr(test_overflow,
test_underflow,
precision=ML_Bool,
tag="std_test",
likely=False,
debug=debug_multi)
#Reconstruction
result = hi_part_value * exp_X * poly + hi_part_value * exp_X
result.set_attributes(tag="result", debug=debug_multi)
C0 = Constant(0, precision=self.precision)
if inline_select:
scheme = Select(
test_std,
Select(test_overflow, FP_PlusInfty(self.precision),
Select(
test_subnormal,
subnormal_result,
C0,
)),
result,
)
return scheme
else:
return_inf = Return(FP_PlusInfty(self.precision))
return_C0 = Return(C0)
return_sub = Return(subnormal_result)
return_std = Return(result)
non_std_statement = Statement(
ConditionBlock(
test_overflow, return_inf,
ConditionBlock(test_subnormal, return_sub, return_C0)))
scheme = Statement(
ConditionBlock(test_std, non_std_statement, return_std))
return scheme
def generate_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_scheme(self):
## convert @p value from an input floating-point precision
# @p in_precision to an output support format @p out_precision
io_precision = self.precision
# declaring main input variable
vx = self.implementation.add_input_signal("x", io_precision)
# rounding mode input
rnd_mode = self.implementation.add_input_signal(
"rnd_mode", rnd_mode_format)
# size of most significant table index (for linear slope tabulation)
alpha = self.alpha # 6
# size of medium significant table index (for initial value table index LSB)
beta = self.beta # 5
# size of least significant table index (for linear offset tabulation)
gamma = self.gamma # 5
guard_bits = self.guard_bits # 3
vx.set_interval(self.interval)
range_hi = sollya.sup(self.interval)
range_lo = sollya.inf(self.interval)
f_hi = self.function(range_hi)
f_lo = self.function(range_lo)
# fixed by format used for reduced_x
range_size = range_hi - range_lo
range_size_log2 = int(sollya.log2(range_size))
assert 2**range_size_log2 == range_size
print("range_size_log2={}".format(range_size_log2))
reduced_x = Conversion(BitLogicRightShift(vx - range_lo,
range_size_log2),
precision=fixed_point(0,
alpha + beta + gamma,
signed=False),
tag="reduced_x",
debug=debug_fixed)
alpha_index = get_fixed_slice(reduced_x,
0,
alpha - 1,
align_hi=FixedPointPosition.FromMSBToLSB,
align_lo=FixedPointPosition.FromMSBToLSB,
tag="alpha_index",
debug=debug_std)
gamma_index = get_fixed_slice(reduced_x,
gamma - 1,
0,
align_hi=FixedPointPosition.FromLSBToLSB,
align_lo=FixedPointPosition.FromLSBToLSB,
tag="gamma_index",
debug=debug_std)
beta_index = get_fixed_slice(reduced_x,
alpha,
gamma,
align_hi=FixedPointPosition.FromMSBToLSB,
align_lo=FixedPointPosition.FromLSBToLSB,
tag="beta_index",
debug=debug_std)
# Assuming monotonic function
f_absmax = max(abs(f_hi), abs(f_lo))
f_absmin = min(abs(f_hi), abs(f_lo))
f_msb = int(sollya.ceil(sollya.log2(f_absmax))) + 1
f_lsb = int(sollya.floor(sollya.log2(f_absmin)))
storage_lsb = f_lsb - io_precision.get_bit_size() - guard_bits
f_int_size = f_msb
f_frac_size = -storage_lsb
storage_format = fixed_point(f_int_size, f_frac_size, signed=False)
Log.report(Log.Info, "storage_format is {}".format(storage_format))
# table of initial value index
tiv_index = Concatenation(alpha_index,
beta_index,
tag="tiv_index",
debug=debug_std)
# table of offset value index
to_index = Concatenation(alpha_index,
gamma_index,
tag="to_index",
debug=debug_std)
tiv_index_size = alpha + beta
to_index_size = alpha + gamma
Log.report(Log.Info, "initial table structures")
table_iv = ML_NewTable(dimensions=[2**tiv_index_size],
storage_precision=storage_format,
tag="tiv")
table_offset = ML_NewTable(dimensions=[2**to_index_size],
storage_precision=storage_format,
tag="to")
slope_table = [None] * (2**alpha)
slope_delta = 1.0 / sollya.SollyaObject(2**alpha)
delta_u = range_size * slope_delta * 2**-15
Log.report(Log.Info, "computing slope value")
for i in range(2**alpha):
# slope is computed at the middle of range_size interval
slope_x = range_lo + (i + 0.5) * range_size * slope_delta
# TODO: gross approximation of derivatives
f_xpu = self.function(slope_x + delta_u / 2)
f_xmu = self.function(slope_x - delta_u / 2)
slope = (f_xpu - f_xmu) / delta_u
slope_table[i] = slope
range_rcp_steps = 1.0 / sollya.SollyaObject(2**tiv_index_size)
Log.report(Log.Info, "computing value for initial-value table")
for i in range(2**tiv_index_size):
slope_index = i / 2**beta
iv_x = range_lo + i * range_rcp_steps * range_size
offset_x = 0.5 * range_rcp_steps * range_size
# initial value is computed so that the piecewise linear
# approximation intersects the function at iv_x + offset_x
iv_y = self.function(
iv_x + offset_x) - offset_x * slope_table[int(slope_index)]
initial_value = storage_format.round_sollya_object(iv_y)
table_iv[i] = initial_value
# determining table of initial value interval
tiv_min = table_iv[0]
tiv_max = table_iv[0]
for i in range(1, 2**tiv_index_size):
tiv_min = min(tiv_min, table_iv[i])
tiv_max = max(tiv_max, table_iv[i])
table_iv.set_interval(Interval(tiv_min, tiv_max))
offset_step = range_size / S2**(alpha + beta + gamma)
for i in range(2**alpha):
Log.report(Log.Info,
"computing offset value for sub-table {}".format(i))
for j in range(2**gamma):
to_i = i * 2**gamma + j
offset = slope_table[i] * j * offset_step
table_offset[to_i] = offset
# determining table of offset interval
to_min = table_offset[0]
to_max = table_offset[0]
for i in range(1, 2**(alpha + gamma)):
to_min = min(to_min, table_offset[i])
to_max = max(to_max, table_offset[i])
offset_interval = Interval(to_min, to_max)
table_offset.set_interval(offset_interval)
initial_value = TableLoad(table_iv,
tiv_index,
precision=storage_format,
tag="initial_value",
debug=debug_fixed)
offset_precision = get_fixed_type_from_interval(offset_interval, 16)
print("offset_precision is {} ({} bits)".format(
offset_precision, offset_precision.get_bit_size()))
table_offset.get_precision().storage_precision = offset_precision
# rounding table value
for i in range(1, 2**(alpha + gamma)):
table_offset[i] = offset_precision.round_sollya_object(
table_offset[i])
offset_value = TableLoad(table_offset,
to_index,
precision=offset_precision,
tag="offset_value",
debug=debug_fixed)
Log.report(
Log.Verbose,
"initial_value's interval: {}, offset_value's interval: {}".format(
evaluate_range(initial_value), evaluate_range(offset_value)))
final_add = initial_value + offset_value
round_bit = final_add # + FixedPointPosition(final_add, io_precision.get_bit_size(), align=FixedPointPosition.FromMSBToLSB)
vr_out = Conversion(initial_value + offset_value,
precision=io_precision,
tag="vr_out",
debug=debug_fixed)
self.implementation.add_output_signal("vr_out", vr_out)
# Approximation error evaluation
approx_error = 0.0
for i in range(2**alpha):
for j in range(2**beta):
tiv_i = (i * 2**beta + j)
# = range_lo + tiv_i * range_rcp_steps * range_size
iv = table_iv[tiv_i]
for k in range(2**gamma):
to_i = i * 2**gamma + k
offset = table_offset[to_i]
approx_value = offset + iv
table_x = range_lo + range_size * (
(i * 2**beta + j) * 2**gamma + k) / S2**(alpha + beta +
gamma)
local_error = abs(1 / (table_x) - approx_value)
approx_error = max(approx_error, local_error)
error_log2 = float(sollya.log2(approx_error))
print("approx_error is {}, error_log2 is {}".format(
float(approx_error), error_log2))
# table size
table_iv_size = 2**(alpha + beta)
table_offset_size = 2**(alpha + gamma)
print("tables' size are {} entries".format(table_iv_size +
table_offset_size))
return [self.implementation]
def generate_scheme(self):
memory_limit = 2500
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = input_var
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
### Constants computations ###
v_log2_hi = nearestint(log(2) * 2**-52) * 2**52
v_log2_lo = round(log(2) - v_log2_hi, 64 + 53, 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")
print "\n\033[1mSearch parameters for the argument reduction:\033[0m (this can take a while)"
arg_reduc = self.generate_argument_reduction(memory_limit)
print "\n\033[1mArgument reduction found:\033[0m [({},{}),({},{})] -> polynomials of degree {},{}, using {} bytes of memory".format(
arg_reduc['size1'], arg_reduc['prec1'], arg_reduc['size2'],
arg_reduc['prec2'], arg_reduc['degree_poly1'],
arg_reduc['degree_poly2'], arg_reduc['sizeof_tables'])
print "\n\033[1mGenerate the first logarithm table:\033[0m containing {} elements, using {} bytes of memory".format(
arg_reduc['length_table1'], arg_reduc['sizeof_table1'])
inv_table_1 = ML_NewTable(
dimensions=[arg_reduc['length_table1']],
storage_precision=ML_Custom_FixedPoint_Format(
1, arg_reduc['prec1'], False),
tag=self.uniquify_name("inv_table_1"))
log_table_1 = ML_NewTable(
dimensions=[arg_reduc['length_table1']],
storage_precision=ML_Custom_FixedPoint_Format(11, 128 - 11, False),
tag=self.uniquify_name("log_table_1"))
for i in range(0, arg_reduc['length_table1'] - 1):
x1 = 1 + i / S2 * arg_reduc['size1']
inv_x1 = ceil(S2**arg_reduc['prec1'] / x1) * S2**arg_reduc['prec1']
log_x1 = floor(log(x1) * S2**(128 - 11)) * S2**(11 - 128)
inv_table_1[
i] = inv_x1 #Constant(inv_x1, precision = ML_Custom_FixedPoint_Format(1, arg_reduc['prec1'], False))
log_table_1[
i] = log_x1 #Constant(log_x1, precision = ML_Custom_FixedPoint_Format(11, 128-11, False))
print "\n\033[1mGenerate the second logarithm table:\033[0m containing {} elements, using {} bytes of memory".format(
arg_reduc['length_table2'], arg_reduc['sizeof_table2'])
inv_table_2 = ML_NewTable(
dimensions=[arg_reduc['length_table2']],
storage_precision=ML_Custom_FixedPoint_Format(
1, arg_reduc['prec2'], False),
tag=self.uniquify_name("inv_table_2"))
log_table_2 = ML_NewTable(
dimensions=[arg_reduc['length_table2']],
storage_precision=ML_Custom_FixedPoint_Format(11, 128 - 11, False),
tag=self.uniquify_name("log_table_2"))
for i in range(0, arg_reduc['length_table2'] - 1):
y1 = 1 + i / S2**arg_reduc['size2']
inv_y1 = ceil(S2**arg_reduc['prec2'] / x1) * S2**arg_reduc['prec2']
log_y1 = floor(log(inv_y1) * S2**(128 - 11)) * S2**(11 - 128)
inv_table_2[
i] = inv_y1 #Constant(inv_y1, precision = ML_Custom_FixedPoint_Format(1, arg_reduc['prec2'], False))
log_table_2[
i] = log_y1 #Constant(log_y1, precision = ML_Custom_FixedPoint_Format(11, 128-11, False))
### Evaluation Scheme ###
print "\n\033[1mGenerate the evaluation scheme:\033[0m"
input_var = self.implementation.add_input_variable(
"input_var", self.precision)
ve = ExponentExtraction(input_var, tag="x_exponent", debug=debugd)
vx = MantissaExtraction(input_var,
tag="x_mantissa",
precision=ML_Custom_FixedPoint_Format(
0, 52, False),
debug=debug_lftolx)
#vx = MantissaExtraction(input_var, tag = "x_mantissa", precision = self.precision, debug = debug_lftolx)
print "filtering and handling special cases"
test_is_special_cases = LogicalNot(
Test(input_var,
specifier=Test.IsIEEENormalPositive,
likely=True,
debug=debugd,
tag="is_special_cases"))
handling_special_cases = Statement(
ConditionBlock(
Test(input_var, specifier=Test.IsSignalingNaN, debug=True),
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision))),
ConditionBlock(Test(input_var, specifier=Test.IsNaN, debug=True),
Return(input_var)) #,
# TODO: add tests for x == 0 (raise DivideByZero, return -Inf), x < 0 (raise InvalidOperation, return qNaN)
# all that remains is x is a subnormal positive
#Statement(
# ReferenceAssign(Dereference(ve), Subtraction(ve, Subtraction(CountLeadingZeros(input_var, tag = 'subnormal_clz', precision = ve.get_precision()), Constant(12, precision = ve.get_precision())))),
# ReferenceAssign(Dereference(vx), BitLogicLeftShift(vx, Addition(CountLeadingZeros(input_var, tag = 'subnormal_clz', precision = ve.get_precision()), Constant(1, precision = ve.get_precision()))))
#)
)
print "doing the argument reduction"
v_dx = vx
v_x1 = Conversion(v_dx,
tag='x1',
precision=ML_Custom_FixedPoint_Format(
0, arg_reduc['size1'], False),
rounding_mode=ML_RoundTowardMinusInfty)
v_index_x = TypeCast(
v_x1, tag='index_x', precision=ML_Int32
) #ML_Custom_FixedPoint_Format(v_x1.get_precision().get_c_bit_size(), 0, False))
v_inv_x = TableLoad(inv_table_1, v_index_x, tag='inv_x')
v_x = Addition(v_dx,
1,
tag='x',
precision=ML_Custom_FixedPoint_Format(1, 52, False))
v_dy = Multiplication(v_x,
v_inv_x,
tag='dy',
precision=ML_Custom_FixedPoint_Format(
0, 52 + arg_reduc['prec1'], False))
v_y1 = Conversion(v_dy,
tag='y1',
precision=ML_Custom_FixedPoint_Format(
0, arg_reduc['size2'], False),
rounding_mode=ML_RoundTowardMinusInfty)
v_index_y = TypeCast(
v_y1, tag='index_y', precision=ML_Int32
) #ML_Custom_FixedPoint_Format(v_y1.get_precision().get_c_bit_size(), 0, False))
v_inv_y = TableLoad(inv_table_2, v_index_y, tag='inv_y')
v_y = Addition(v_dy,
1,
tag='y',
precision=ML_Custom_FixedPoint_Format(
1, 52 + arg_reduc['prec2'], False))
# note that we limit the number of bits used to represent dz to 64.
# we proved during the arg reduction that we can do that (sup(out_interval) < 2^(64-52-prec1-prec2))
v_dz = Multiplication(
v_y,
v_inv_y,
tag='z',
precision=ML_Custom_FixedPoint_Format(
64 - 52 - arg_reduc['prec1'] - arg_reduc['prec2'],
52 + arg_reduc['prec1'] + arg_reduc['prec2'], False))
# reduce the number of bits used to represent dz. we can do that
print "doing the first polynomial evaluation"
global_poly1_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, arg_reduc['degree_poly1'] - 1,
[64] * (arg_reduc['degree_poly1']), arg_reduc['out_interval'],
fixed, sollya.absolute)
poly1_object = global_poly1_object.sub_poly(start_index=1)
print global_poly1_object
print poly1_object
poly1 = PolynomialSchemeEvaluator.generate_horner_scheme(
poly1_object, v_dz, unified_precision=v_dz.get_precision())
return ConditionBlock(test_is_special_cases, handling_special_cases,
Return(poly1))
#approx_interval = Interval(0, 27021597764222975*S2**-61)
#poly_degree = 1+sup(guessdegree(log(1+x)/x, approx_interval, S2**-(self.precision.get_field_size())))
#global_poly_object = Polynomial.build_from_approximation(log(1+x)/x, poly_degree, [1] + [self.precision]*(poly_degree), approx_interval, sollya.absolute)
#poly_object = global_poly_object.sub_poly(start_index = 1)
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
#_poly.set_attributes(tag = "poly", debug = debug_lftolx)
"""
def generate_scheme(self):
""" generate scheme """
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
# retrieving processor inverse approximation table
lo_bound_global = SollyaObject(0.0)
hi_bound_global = SollyaObject(0.75)
approx_interval = Interval(lo_bound_global, hi_bound_global)
approx_interval_size = hi_bound_global - lo_bound_global
# table creation
table_index_size = 7
field_index_size = 2
exp_index_size = table_index_size - field_index_size
table_size = 2**table_index_size
table_index_range = range(table_size)
local_degree = 9
coeff_table = ML_NewTable(dimensions=[table_size, local_degree],
storage_precision=self.precision)
exp_lo = 2**exp_index_size
for i in table_index_range:
lo_bound = (1.0 + (i % 2**field_index_size) * S2**-field_index_size
) * S2**(i / 2**field_index_size - exp_lo)
hi_bound = (1.0 +
((i % 2**field_index_size) + 1) * S2**-field_index_size
) * S2**(i / 2**field_index_size - exp_lo)
local_approx_interval = Interval(lo_bound, hi_bound)
local_poly_object, local_error = Polynomial.build_from_approximation_with_error(
acos(1 - sollya.x), local_degree,
[self.precision] * (local_degree + 1), local_approx_interval,
sollya.absolute)
local_error = int(
log2(sup(abs(local_error / acos(1 - local_approx_interval)))))
coeff_table
for d in range(local_degree):
coeff_table[i][d] = sollya.coeff(
local_poly_object.get_sollya_object(), d)
table_index = BitLogicRightShift(
vx,
vx.get_precision().get_field_size() -
field_index_size) - (exp_lo << field_index_size)
print "building mathematical polynomial"
poly_degree = sup(
sollya.guessdegree(acos(x), approx_interval,
S2**-(self.precision.get_field_size())))
print "guessed polynomial degree: ", int(poly_degree)
#global_poly_object = Polynomial.build_from_approximation(log10(1+x)/x, poly_degree, [self.precision]*(poly_degree+1), approx_interval, absolute)
print "generating polynomial evaluation scheme"
#_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, _red_vx, unified_precision = self.precision)
# building eval error map
#eval_error_map = {
# red_vx: Variable("red_vx", precision = self.precision, interval = red_vx.get_interval()),
# log_inv_hi: Variable("log_inv_hi", precision = self.precision, interval = table_high_interval),
# log_inv_lo: Variable("log_inv_lo", precision = self.precision, interval = table_low_interval),
#}
# computing gappa error
#poly_eval_error = self.get_eval_error(result, eval_error_map)
# main scheme
print "MDL scheme"
scheme = Statement(Return(vx))
return scheme
def generate_payne_hanek(vx,
frac_pi,
precision,
n=100,
k=4,
chunk_num=None,
debug=False):
""" generate payne and hanek argument reduction for frac_pi * variable """
sollya.roundingwarnings = sollya.off
debug_precision = debug_multi
int_precision = {ML_Binary32: ML_Int32, ML_Binary64: ML_Int64}[precision]
p = precision.get_field_size()
# weight of the most significant digit of the constant
cst_msb = floor(log2(abs(frac_pi)))
# length of exponent range which must be covered by the approximation
# of the constant
cst_exp_range = cst_msb - precision.get_emin_subnormal() + 1
# chunk size has to be so than multiplication by a splitted <v>
# (vx_hi or vx_lo) is exact
chunk_size = precision.get_field_size() / 2 - 2
chunk_number = int(ceil((cst_exp_range + chunk_size - 1) / chunk_size))
scaling_factor = S2**-(chunk_size / 2)
chunk_size_cst = Constant(chunk_size, precision=ML_Int32)
cst_msb_node = Constant(cst_msb, precision=ML_Int32)
# Saving sollya's global precision
old_global_prec = sollya.settings.prec
sollya.settings.prec(cst_exp_range + n)
# table to store chunk of constant multiplicand
cst_table = ML_NewTable(dimensions=[chunk_number, 1],
storage_precision=precision,
tag="PH_cst_table")
# table to store sqrt(scaling_factor) corresponding to the
# cst multiplicand chunks
scale_table = ML_NewTable(dimensions=[chunk_number, 1],
storage_precision=precision,
tag="PH_scale_table")
tmp_cst = frac_pi
# cst_table stores normalized constant chunks (they have been
# scale back to close to 1.0 interval)
#
# scale_table stores the scaling factors corresponding to the
# denormalization of cst_table coefficients
# this loop divide the digits of frac_pi into chunks
# the chunk lsb weight is given by a shift from
# cst_msb, multiple of the chunk index
for i in range(chunk_number):
value_div_factor = S2**(chunk_size * (i + 1) - cst_msb)
local_cst = int(tmp_cst * value_div_factor) / value_div_factor
local_scale = (scaling_factor**i)
# storing scaled constant chunks
cst_table[i][0] = local_cst / (local_scale**2)
scale_table[i][0] = local_scale
# Updating constant value
tmp_cst = tmp_cst - local_cst
# Computing which part of the constant we do not need to multiply
# In the following comments, vi represents the bit of frac_pi of weight 2**-i
# Bits vi so that i <= (vx_exp - p + 1 -k) are not needed, because they result
# in a multiple of 2pi and do not contribute to trig functions.
vx_exp = ExponentExtraction(
vx, precision=vx.get_precision().get_integer_format())
vx_exp = Conversion(vx_exp, precision=ML_Int32)
msb_exp = -(vx_exp - p + 1 - k)
msb_exp.set_attributes(tag="msb_exp", debug=debug_multi)
msb_exp = Conversion(msb_exp, precision=ML_Int32)
# Select the highest index where the reduction should start
msb_index = Select(cst_msb_node < msb_exp, 0,
(cst_msb_node - msb_exp) / chunk_size_cst)
msb_index.set_attributes(tag="msb_index", debug=debug_multi)
# For a desired accuracy of 2**-n, bits vi so that i >= (vx_exp + n + 4) are not needed, because they contribute less than
# 2**-n to the result
lsb_exp = -(vx_exp + n + 4)
lsb_exp.set_attributes(tag="lsb_exp", debug=debug_multi)
lsb_exp = Conversion(lsb_exp, precision=ML_Int32)
# Index of the corresponding chunk
lsb_index = (cst_msb_node - lsb_exp) / chunk_size_cst
lsb_index.set_attributes(tag="lsb_index", debug=debug_multi)
# Splitting vx
half_size = precision.get_field_size() / 2 + 1
# hi part (most significant digit) of vx input
vx_hi = TypeCast(BitLogicAnd(
TypeCast(vx, precision=int_precision),
Constant(~int(2**half_size - 1), precision=int_precision)),
precision=precision)
vx_hi.set_attributes(tag="vx_hi_ph") #, debug = debug_multi)
vx_lo = vx - vx_hi
vx_lo.set_attributes(tag="vx_lo_ph") #, debug = debug_multi)
# loop iterator variable
vi = Variable("i", precision=ML_Int32, var_type=Variable.Local)
# step scaling factor
half_scaling = Constant(S2**(-chunk_size / 2), precision=precision)
i1 = Constant(1, precision=ML_Int32)
# accumulator to the output precision
acc = Variable("acc", precision=precision, var_type=Variable.Local)
# integer accumulator
acc_int = Variable("acc_int",
precision=int_precision,
var_type=Variable.Local)
init_loop = Statement(
vx_hi,
vx_lo,
ReferenceAssign(vi, msb_index),
ReferenceAssign(acc, Constant(0, precision=precision)),
ReferenceAssign(acc_int, Constant(0, precision=int_precision)),
)
cst_load = TableLoad(cst_table,
vi,
0,
tag="cst_load",
debug=debug_precision)
sca_load = TableLoad(scale_table,
vi,
0,
tag="sca_load",
debug=debug_precision)
# loop body
# hi_mult = vx_hi * <scale_factor> * <cst>
hi_mult = (vx_hi * sca_load) * (cst_load * sca_load)
hi_mult.set_attributes(tag="hi_mult", debug=debug_precision)
pre_hi_mult_int = NearestInteger(hi_mult,
precision=int_precision,
tag="hi_mult_int",
debug=(debuglld if debug else None))
hi_mult_int_f = Conversion(pre_hi_mult_int,
precision=precision,
tag="hi_mult_int_f",
debug=debug_precision)
pre_hi_mult_red = (hi_mult - hi_mult_int_f).modify_attributes(
tag="hi_mult_red", debug=debug_precision)
# for the first chunks (vx_hi * <constant chunk>) exceeds 2**k+1 and may be
# discard (whereas it may lead to overflow during integer conversion
pre_exclude_hi = ((cst_msb_node - (vi + i1) * chunk_size + i1) +
(vx_exp + Constant(-half_size + 1, precision=ML_Int32))
).modify_attributes(tag="pre_exclude_hi",
debug=(debugd if debug else None))
pre_exclude_hi.propagate_precision(ML_Int32,
[cst_msb_node, vi, vx_exp, i1])
Ck = Constant(k, precision=ML_Int32)
exclude_hi = pre_exclude_hi <= Ck
exclude_hi.set_attributes(tag="exclude_hi", debug=debug_multi)
hi_mult_red = Select(exclude_hi, pre_hi_mult_red,
Constant(0, precision=precision))
hi_mult_int = Select(exclude_hi, pre_hi_mult_int,
Constant(0, precision=int_precision))
# lo part of the chunk reduction
lo_mult = (vx_lo * sca_load) * (cst_load * sca_load)
lo_mult.set_attributes(tag="lo_mult") #, debug = debug_multi)
lo_mult_int = NearestInteger(lo_mult,
precision=int_precision,
tag="lo_mult_int") #, debug = debug_multi
lo_mult_int_f = Conversion(lo_mult_int,
precision=precision,
tag="lo_mult_int_f") #, debug = debug_multi)
lo_mult_red = (lo_mult - lo_mult_int_f).modify_attributes(
tag="lo_mult_red") #, debug = debug_multi)
# accumulating fractional part
acc_expr = (acc + hi_mult_red) + lo_mult_red
# accumulating integer part
int_expr = ((acc_int + hi_mult_int) + lo_mult_int) % 2**(k + 1)
CF1 = Constant(1, precision=precision)
CI1 = Constant(1, precision=int_precision)
# extracting exceeding integer part in fractionnal accumulator
acc_expr_int = NearestInteger(acc_expr, precision=int_precision)
# normalizing integer and fractionnal accumulator by subtracting then
# adding exceeding integer part
normalization = Statement(
ReferenceAssign(
acc, acc_expr - Conversion(acc_expr_int, precision=precision)),
ReferenceAssign(acc_int, int_expr + acc_expr_int),
)
acc_expr.set_attributes(tag="acc_expr") #, debug = debug_multi)
int_expr.set_attributes(tag="int_expr") #, debug = debug_multi)
red_loop = Loop(
init_loop, vi <= lsb_index,
Statement(acc_expr, int_expr, normalization,
ReferenceAssign(vi, vi + 1)))
result = Statement(lsb_index, msb_index, red_loop)
# restoring sollya's global precision
sollya.settings.prec = old_global_prec
return result, acc, acc_int
def generate_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = self.implementation.add_input_variable("x", self.precision)
Log.report(Log.Info, "generating implementation scheme")
if self.debug_flag:
Log.report(Log.Info, "debug has been enabled")
# local overloading of RaiseReturn operation
def SincosRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
sollya_precision = self.precision.get_sollya_object()
hi_precision = self.precision.get_field_size() - 8
cw_hi_precision = self.precision.get_field_size() - 4
ext_precision = {
ML_Binary32: ML_Binary64,
ML_Binary64: ML_Binary64
}[self.precision]
int_precision = {
ML_Binary32: ML_Int32,
ML_Binary64: ML_Int64
}[self.precision]
if self.precision is ML_Binary32:
ph_bound = S2**10
else:
ph_bound = S2**33
test_ph_bound = Comparison(vx,
ph_bound,
specifier=Comparison.GreaterOrEqual,
precision=ML_Bool,
likely=False)
# argument reduction
# m
frac_pi_index = {ML_Binary32: 10, ML_Binary64: 14}[self.precision]
C0 = Constant(0, precision=int_precision)
C1 = Constant(1, precision=int_precision)
C_offset = Constant(3 * S2**(frac_pi_index - 1),
precision=int_precision)
# 2^m / pi
frac_pi = round(S2**frac_pi_index / pi, cw_hi_precision, sollya.RN)
frac_pi_lo = round(S2**frac_pi_index / pi - frac_pi, sollya_precision,
sollya.RN)
# pi / 2^m, high part
inv_frac_pi = round(pi / S2**frac_pi_index, cw_hi_precision, sollya.RN)
# pi / 2^m, low part
inv_frac_pi_lo = round(pi / S2**frac_pi_index - inv_frac_pi,
sollya_precision, sollya.RN)
# computing k
vx.set_attributes(tag="vx", debug=debug_multi)
vx_pi = Addition(Multiplication(vx,
Constant(frac_pi,
precision=self.precision),
precision=self.precision),
Multiplication(vx,
Constant(frac_pi_lo,
precision=self.precision),
precision=self.precision),
precision=self.precision,
tag="vx_pi",
debug=debug_multi)
k = NearestInteger(vx_pi,
precision=int_precision,
tag="k",
debug=debug_multi)
# k in floating-point precision
fk = Conversion(k,
precision=self.precision,
tag="fk",
debug=debug_multi)
inv_frac_pi_cst = Constant(inv_frac_pi,
tag="inv_frac_pi",
precision=self.precision,
debug=debug_multi)
inv_frac_pi_lo_cst = Constant(inv_frac_pi_lo,
tag="inv_frac_pi_lo",
precision=self.precision,
debug=debug_multi)
# Cody-Waite reduction
red_coeff1 = Multiplication(fk,
inv_frac_pi_cst,
precision=self.precision,
exact=True)
red_coeff2 = Multiplication(Negation(fk, precision=self.precision),
inv_frac_pi_lo_cst,
precision=self.precision,
exact=True)
# Should be exact / Sterbenz' Lemma
pre_sub_mul = Subtraction(vx,
red_coeff1,
precision=self.precision,
exact=True)
# Fast2Sum
s = Addition(pre_sub_mul,
red_coeff2,
precision=self.precision,
unbreakable=True,
tag="s",
debug=debug_multi)
z = Subtraction(s,
pre_sub_mul,
precision=self.precision,
unbreakable=True,
tag="z",
debug=debug_multi)
t = Subtraction(red_coeff2,
z,
precision=self.precision,
unbreakable=True,
tag="t",
debug=debug_multi)
red_vx_std = Addition(s, t, precision=self.precision)
red_vx_std.set_attributes(tag="red_vx_std", debug=debug_multi)
# To compute sine we offset x by 3pi/2
# which means add 3 * S2^(frac_pi_index-1) to k
if self.sin_output:
Log.report(Log.Info, "Computing Sin")
offset_k = Addition(k,
C_offset,
precision=int_precision,
tag="offset_k")
else:
Log.report(Log.Info, "Computing Cos")
offset_k = k
modk = Variable("modk",
precision=int_precision,
var_type=Variable.Local)
red_vx = Variable("red_vx",
precision=self.precision,
var_type=Variable.Local)
# Faster modulo using bitwise logic
modk_std = BitLogicAnd(offset_k,
2**(frac_pi_index + 1) - 1,
precision=int_precision,
tag="modk",
debug=debug_multi)
approx_interval = Interval(-pi / (S2**(frac_pi_index + 1)),
pi / S2**(frac_pi_index + 1))
red_vx.set_interval(approx_interval)
Log.report(Log.Info, "approx interval: %s\n" % approx_interval)
Log.report(Log.Info,
"building tabulated approximation for sin and cos")
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
# polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
table_index_size = frac_pi_index + 1
cos_table = ML_NewTable(dimensions=[2**table_index_size, 1],
storage_precision=self.precision,
tag=self.uniquify_name("cos_table"))
for i in range(2**(frac_pi_index + 1)):
local_x = i * pi / S2**frac_pi_index
cos_local = round(cos(local_x), self.precision.get_sollya_object(),
sollya.RN)
cos_table[i][0] = cos_local
sin_index = Modulo(modk + 2**(frac_pi_index - 1),
2**(frac_pi_index + 1),
precision=int_precision,
tag="sin_index") #, debug = debug_multi)
tabulated_cos = TableLoad(cos_table,
modk,
C0,
precision=self.precision,
tag="tab_cos",
debug=debug_multi)
tabulated_sin = -TableLoad(cos_table,
sin_index,
C0,
precision=self.precision,
tag="tab_sin",
debug=debug_multi)
poly_degree_cos = sup(
guessdegree(cos(sollya.x), approx_interval, S2**
-self.precision.get_precision()) + 2)
poly_degree_sin = sup(
guessdegree(
sin(sollya.x) / sollya.x, approx_interval, S2**
-self.precision.get_precision()) + 2)
poly_degree_cos_list = range(0, int(poly_degree_cos) + 3)
poly_degree_sin_list = range(0, int(poly_degree_sin) + 3)
# cosine polynomial: limiting first and second coefficient precision to 1-bit
poly_cos_prec_list = [self.precision] * len(poly_degree_cos_list)
# sine polynomial: limiting first coefficient precision to 1-bit
poly_sin_prec_list = [self.precision] * len(poly_degree_sin_list)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
Log.report(Log.Info,
"building mathematical polynomials for sin and cos")
# Polynomial approximations
Log.report(Log.Info, "cos")
poly_object_cos, poly_error_cos = Polynomial.build_from_approximation_with_error(
cos(sollya.x),
poly_degree_cos_list,
poly_cos_prec_list,
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(Log.Info, "sin")
poly_object_sin, poly_error_sin = Polynomial.build_from_approximation_with_error(
sin(sollya.x),
poly_degree_sin_list,
poly_sin_prec_list,
approx_interval,
sollya.absolute,
error_function=error_function)
Log.report(
Log.Info, "poly error cos: {} / {:d}".format(
poly_error_cos, int(sollya.log2(poly_error_cos))))
Log.report(
Log.Info, "poly error sin: {0} / {1:d}".format(
poly_error_sin, int(sollya.log2(poly_error_sin))))
Log.report(Log.Info, "poly cos : %s" % poly_object_cos)
Log.report(Log.Info, "poly sin : %s" % poly_object_sin)
# Polynomial evaluation scheme
poly_cos = polynomial_scheme_builder(
poly_object_cos.sub_poly(start_index=1),
red_vx,
unified_precision=self.precision)
poly_sin = polynomial_scheme_builder(
poly_object_sin.sub_poly(start_index=2),
red_vx,
unified_precision=self.precision)
poly_cos.set_attributes(tag="poly_cos", debug=debug_multi)
poly_sin.set_attributes(tag="poly_sin",
debug=debug_multi,
unbreakable=True)
# TwoProductFMA
mul_cos_x = tabulated_cos * poly_cos
mul_cos_y = FusedMultiplyAdd(tabulated_cos,
poly_cos,
-mul_cos_x,
precision=self.precision)
mul_sin_x = tabulated_sin * poly_sin
mul_sin_y = FusedMultiplyAdd(tabulated_sin,
poly_sin,
-mul_sin_x,
precision=self.precision)
mul_coeff_sin_hi = tabulated_sin * red_vx
mul_coeff_sin_lo = FusedMultiplyAdd(tabulated_sin, red_vx,
-mul_coeff_sin_hi)
mul_cos = Addition(mul_cos_x,
mul_cos_y,
precision=self.precision,
tag="mul_cos") #, debug = debug_multi)
mul_sin = Negation(Addition(mul_sin_x,
mul_sin_y,
precision=self.precision),
precision=self.precision,
tag="mul_sin") #, debug = debug_multi)
mul_coeff_sin = Negation(Addition(mul_coeff_sin_hi,
mul_coeff_sin_lo,
precision=self.precision),
precision=self.precision,
tag="mul_coeff_sin") #, debug = debug_multi)
mul_cos_x.set_attributes(
tag="mul_cos_x", precision=self.precision) #, debug = debug_multi)
mul_cos_y.set_attributes(
tag="mul_cos_y", precision=self.precision) #, debug = debug_multi)
mul_sin_x.set_attributes(
tag="mul_sin_x", precision=self.precision) #, debug = debug_multi)
mul_sin_y.set_attributes(
tag="mul_sin_y", precision=self.precision) #, debug = debug_multi)
cos_eval_d_1 = (((mul_cos + mul_sin) + mul_coeff_sin) + tabulated_cos)
cos_eval_d_1.set_attributes(tag="cos_eval_d_1",
precision=self.precision,
debug=debug_multi)
result_1 = Statement(Return(cos_eval_d_1))
#######################################################################
# LARGE ARGUMENT MANAGEMENT #
# (lar: Large Argument Reduction) #
#######################################################################
# payne and hanek argument reduction for large arguments
ph_k = frac_pi_index
ph_frac_pi = round(S2**ph_k / pi, 1500, sollya.RN)
ph_inv_frac_pi = pi / S2**ph_k
ph_statement, ph_acc, ph_acc_int = generate_payne_hanek(vx,
ph_frac_pi,
self.precision,
n=100,
k=ph_k)
# assigning Large Argument Reduction reduced variable
lar_vx = Variable("lar_vx",
precision=self.precision,
var_type=Variable.Local)
lar_red_vx = Addition(Multiplication(lar_vx,
inv_frac_pi,
precision=self.precision),
Multiplication(lar_vx,
inv_frac_pi_lo,
precision=self.precision),
precision=self.precision,
tag="lar_red_vx",
debug=debug_multi)
C32 = Constant(2**(ph_k + 1), precision=int_precision, tag="C32")
ph_acc_int_red = Select(ph_acc_int < C0,
C32 + ph_acc_int,
ph_acc_int,
precision=int_precision,
tag="ph_acc_int_red")
if self.sin_output:
lar_offset_k = Addition(ph_acc_int_red,
C_offset,
precision=int_precision,
tag="lar_offset_k")
else:
lar_offset_k = ph_acc_int_red
ph_acc_int_red.set_attributes(tag="ph_acc_int_red", debug=debug_multi)
lar_modk = BitLogicAnd(lar_offset_k,
2**(frac_pi_index + 1) - 1,
precision=int_precision,
tag="lar_modk",
debug=debug_multi)
lar_statement = Statement(ph_statement,
ReferenceAssign(lar_vx,
ph_acc,
debug=debug_multi),
ReferenceAssign(red_vx,
lar_red_vx,
debug=debug_multi),
ReferenceAssign(modk, lar_modk),
prevent_optimization=True)
test_NaN_or_Inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
tag="NaN_or_Inf",
debug=debug_multi)
return_NaN_or_Inf = Statement(Return(FP_QNaN(self.precision)))
scheme = ConditionBlock(
test_NaN_or_Inf, Statement(ClearException(), return_NaN_or_Inf),
Statement(
modk, red_vx,
ConditionBlock(
test_ph_bound, lar_statement,
Statement(
ReferenceAssign(modk, modk_std),
ReferenceAssign(red_vx, red_vx_std),
)), result_1))
return scheme
def piecewise_approximation(function,
variable,
precision,
bound_low=-1.0,
bound_high=1.0,
num_intervals=16,
max_degree=2,
error_threshold=S2**-24,
odd=False,
even=False):
""" Generate a piecewise approximation
:param function: function to be approximated
:type function: SollyaObject
:param variable: input variable
:type variable: Variable
:param precision: variable's format
:type precision: ML_Format
:param bound_low: lower bound for the approximation interval
:param bound_high: upper bound for the approximation interval
:param num_intervals: number of sub-interval / sub-division of the main interval
:param max_degree: maximum degree for an approximation on any sub-interval
:param error_threshold: error bound for an approximation on any sub-interval
:return: pair (scheme, error) where scheme is a graph node for an
approximation scheme of function evaluated at variable, and error
is the maximum approximation error encountered
:rtype tuple(ML_Operation, SollyaObject): """
degree_generator = piecewise_approximation_degree_generator(
function,
bound_low,
bound_high,
num_intervals=num_intervals,
error_threshold=error_threshold,
)
degree_list = list(degree_generator)
# if max_degree is None then we determine it locally
if max_degree is None:
max_degree = max(degree_list)
# table to store coefficients of the approximation on each segment
coeff_table = ML_NewTable(
dimensions=[num_intervals, max_degree + 1],
storage_precision=precision,
tag="coeff_table",
const=True # by default all approximation coeff table are const
)
error_function = lambda p, f, ai, mod, t: sollya.dirtyinfnorm(p - f, ai)
max_approx_error = 0.0
interval_size = (bound_high - bound_low) / num_intervals
for i in range(num_intervals):
subint_low = bound_low + i * interval_size
subint_high = bound_low + (i + 1) * interval_size
local_function = function(sollya.x + subint_low)
local_interval = Interval(-interval_size, interval_size)
local_degree = degree_list[i]
if local_degree > max_degree:
Log.report(
Log.Warning,
"local_degree {} exceeds max_degree bound ({}) in piecewise_approximation",
local_degree, max_degree)
# as max_degree defines the size of the table we can use
# it as the degree for each sub-interval polynomial
# as there is nothing to gain (yet) by using a smaller polynomial
degree = max_degree # min(max_degree, local_degree)
if function(subint_low) == 0.0:
# if the lower bound is a zero to the function, we
# need to force value=0 for the constant coefficient
# and extend the approximation interval
local_poly_degree_list = list(
range(1 if even else 0, degree + 1, 2 if odd or even else 1))
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
function(sollya.x) / sollya.x,
local_poly_degree_list,
[precision] * len(local_poly_degree_list),
Interval(-subint_high * 0.95, subint_high),
sollya.absolute,
error_function=error_function)
# multiply by sollya.x
poly_object = poly_object.sub_poly(offset=-1)
else:
try:
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
local_function,
degree, [precision] * (degree + 1),
local_interval,
sollya.absolute,
error_function=error_function)
except SollyaError as err:
# try to see if function is constant on the interval (possible
# failure cause for fpminmax)
cst_value = precision.round_sollya_object(
function(subint_low), sollya.RN)
accuracy = error_threshold
diff_with_cst_range = sollya.supnorm(cst_value, local_function,
local_interval,
sollya.absolute, accuracy)
diff_with_cst = sup(abs(diff_with_cst_range))
if diff_with_cst < error_threshold:
Log.report(Log.Info, "constant polynomial detected")
poly_object = Polynomial([function(subint_low)] +
[0] * degree)
approx_error = diff_with_cst
else:
Log.report(
Log.error,
"degree: {} for index {}, diff_with_cst={} (vs error_threshold={}) ",
degree,
i,
diff_with_cst,
error_threshold,
error=err)
for ci in range(max_degree + 1):
if ci in poly_object.coeff_map:
coeff_table[i][ci] = poly_object.coeff_map[ci]
else:
coeff_table[i][ci] = 0.0
if approx_error > error_threshold:
Log.report(
Log.Warning,
"piecewise_approximation on index {} exceeds error threshold: {} > {}",
i, approx_error, error_threshold)
max_approx_error = max(max_approx_error, abs(approx_error))
# computing offset
diff = Subtraction(variable,
Constant(bound_low, precision=precision),
tag="diff",
debug=debug_multi,
precision=precision)
int_prec = precision.get_integer_format()
# delta = bound_high - bound_low
delta_ratio = Constant(num_intervals / (bound_high - bound_low),
precision=precision)
# computing table index
# index = nearestint(diff / delta * <num_intervals>)
index = Max(0,
Min(
NearestInteger(
Multiplication(diff, delta_ratio, precision=precision),
precision=int_prec,
), num_intervals - 1),
tag="index",
debug=debug_multi,
precision=int_prec)
poly_var = Subtraction(diff,
Multiplication(
Conversion(index, precision=precision),
Constant(interval_size, precision=precision)),
precision=precision,
tag="poly_var",
debug=debug_multi)
# generating indexed polynomial
coeffs = [(ci, TableLoad(coeff_table, index, ci))
for ci in range(max_degree + 1)][::-1]
poly_scheme = PolynomialSchemeEvaluator.generate_horner_scheme2(
coeffs, poly_var, precision, {}, precision)
return poly_scheme, max_approx_error
def generate_scheme(self):
"""Produce an abstract scheme for the logarithm.
This abstract scheme will be used by the code generation backend.
"""
if self.precision not in [ML_Binary32, ML_Binary64]:
Log.report(Log.Error, "The demanded precision is not supported")
vx = self.implementation.add_input_variable("x", self.precision)
def default_bool_convert(optree, precision=None, **kw):
return bool_convert(optree, precision, -1, 0, **kw) \
if isinstance(self.processor, VectorBackend) \
else bool_convert(optree, precision, 1, 0, **kw)
precision = self.precision.sollya_object
int_prec = self.precision.get_integer_format()
Log.report(Log.Info, "int_prec is %s" % int_prec)
uint_prec = self.precision.get_unsigned_integer_format()
Log.report(Log.Info, "MDL constants")
cgpe_scheme_idx = int(self.cgpe_index)
table_index_size = int(self.tbl_index_size)
#
table_nb_elements = 2**(table_index_size)
table_dimensions = [2*table_nb_elements] # two values are stored for each element
field_size = Constant(self.precision.get_field_size(),
precision = int_prec,
tag = 'field_size')
if self.log_radix == EXP_1:
log2_hi = Constant(
round(log(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log(2) - round(log(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
elif self.log_radix == 10:
log2_hi = Constant(
round(log10(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log10(2) - round(log10(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
# ... if log_radix == '2' then log2(2) == 1
# subnormal_mask aims at trapping positive subnormals except zero.
# That's why we will subtract 1 to the integer bitstring of the input, and
# then compare for Less (strict) the resulting integer bitstring to this
# mask, e.g. 0x7fffff for binary32.
if self.no_subnormal == False:
subnormal_mask = Constant((1 << self.precision.get_field_size()) - 1,
precision = int_prec, tag = 'subnormal_mask')
fp_one = Constant(1.0, precision = self.precision, tag = 'fp_one')
fp_one_as_uint = TypeCast(fp_one, precision = uint_prec,
tag = 'fp_one_as_uint')
int_zero = Constant(0, precision = int_prec, tag = 'int_zero')
int_one = Constant(1, precision = int_prec, tag = 'int_one')
table_mantissa_half_ulp = Constant(
1 << (self.precision.field_size - table_index_size - 1),
precision = int_prec
)
table_s_exp_index_mask = Constant(
~((table_mantissa_half_ulp.get_value() << 1) - 1),
precision = uint_prec
)
Log.report(Log.Info, "MDL table")
# The table holds approximations of -log(2^tau * r_i) so we first compute
# the index value for which tau changes from 1 to 0.
cut = sqrt(2.)
tau_index_limit = floor(table_nb_elements * (2./cut - 1))
sollya_logtbl = [
(-log1p(float(i) / table_nb_elements)
+ (0 if i <= tau_index_limit else log(2.))) / log(self.log_radix)
for i in range(table_nb_elements)
]
# ...
init_logtbl_hi = [
round(sollya_logtbl[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl_lo = [
round(sollya_logtbl[i] - init_logtbl_hi[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl = [tmp[i] for i in range(len(init_logtbl_hi)) for tmp in [init_logtbl_hi, init_logtbl_lo]]
log1p_table = ML_NewTable(dimensions = table_dimensions,
storage_precision = self.precision,
init_data = init_logtbl,
tag = 'ml_log1p_table')
# ...
if self.no_rcp:
sollya_rcptbl = [
(1/((1+float(i)/table_nb_elements)+2**(-1-int(self.tbl_index_size))))
for i in range(table_nb_elements)
]
init_rcptbl = [
round(sollya_rcptbl[i],
int(self.tbl_index_size)+1, # self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
rcp_table = ML_NewTable(dimensions = [table_nb_elements],
storage_precision = self.precision,
init_data = init_rcptbl,
tag = 'ml_rcp_table')
# ...
Log.report(Log.Info, 'MDL unified subnormal handling')
vx_as_int = TypeCast(vx, precision = int_prec, tag = 'vx_as_int')
if self.no_subnormal == False:
vx_as_uint = TypeCast(vx, precision = uint_prec, tag = 'vx_as_uint')
# Avoid the 0.0 case by subtracting 1 from vx_as_int
tmp = Comparison(vx_as_int - 1, subnormal_mask,
specifier = Comparison.Less)
is_subnormal = default_bool_convert(
tmp, # Will catch negative values as well as NaNs with sign bit set
precision = int_prec)
is_subnormal.set_attributes(tag = "is_subnormal")
if not(isinstance(self.processor, VectorBackend)):
is_subnormal = Subtraction(Constant(0, precision = int_prec),
is_subnormal,
precision = int_prec)
#################################################
# Vectorizable integer based subnormal handling #
#################################################
# 1. lzcnt
# custom lzcount-like for subnormal numbers using FPU (see draft article)
Zi = BitLogicOr(vx_as_uint, fp_one_as_uint, precision = uint_prec, tag="Zi")
Zf = Subtraction(
TypeCast(Zi, precision = self.precision),
fp_one,
precision = self.precision,
tag="Zf")
# Zf exponent is -(nlz(x) - exponent_size).
# 2. compute shift value
# Vectorial comparison on x86+sse/avx is going to look like
# '|0x00|0xff|0x00|0x00|' and that's why we use Negate.
# But for scalar code generation, comparison will rather be either 0 or 1
# in C. Thus mask below won't be correct for a scalar implementation.
# FIXME: Can we know the backend that will be called and choose in
# consequence? Should we make something arch-agnostic instead?
#
n_value = BitLogicAnd(
Addition(
DirtyExponentExtraction(Zf, self.precision),
Constant(
self.precision.get_bias(),
precision = int_prec),
precision = int_prec),
is_subnormal,
precision = int_prec,
tag = "n_value")
alpha = Negation(n_value, tag="alpha")
#
# 3. shift left
# renormalized_mantissa = BitLogicLeftShift(vx_as_int, value)
normal_vx_as_int = BitLogicLeftShift(vx_as_int, alpha)
# 4. set exponent to the right value
# Compute the exponent to add : (p-1)-(value) + 1 = p-1-value
# The final "+ 1" comes from the fact that once renormalized, the
# floating-point datum has a biased exponent of 1
#tmp0 = Subtraction(
# field_size,
# value,
# precision = int_prec,
# tag="tmp0")
# Set the value to 0 if the number is not subnormal
#tmp1 = BitLogicAnd(tmp0, is_subnormal)
#renormalized_exponent = BitLogicLeftShift(
# tmp1,
# field_size
# )
else: # no_subnormal == True
normal_vx_as_int = vx_as_int
#normal_vx_as_int = renormalized_mantissa + renormalized_exponent
normal_vx = TypeCast(normal_vx_as_int, precision = self.precision,
tag = 'normal_vx')
# alpha = BitLogicAnd(field_size, is_subnormal, tag = 'alpha')
# XXX Extract the mantissa, see if this is supported in the x86 vector
# backend or if it still uses the support_lib.
vx_mantissa = MantissaExtraction(normal_vx, precision = self.precision)
Log.report(Log.Info, "MDL scheme")
if self.force_division == True:
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
elif self.no_rcp == False:
rcp_m = ReciprocalSeed(vx_mantissa, precision = self.precision)
if not self.processor.is_supported_operation(rcp_m):
if self.precision == ML_Binary64:
# Try using a binary32 FastReciprocal
binary32_m = Conversion(vx_mantissa, precision = ML_Binary32)
rcp_m = ReciprocalSeed(binary32_m, precision = ML_Binary32)
rcp_m = Conversion(rcp_m, precision = ML_Binary64)
if not self.processor.is_supported_operation(rcp_m):
# FIXME An approximation table could be used instead but for vector
# implementations another GATHER would be required.
# However this may well be better than a division...
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
else: # ... use a look-up table
rcp_shift = BitLogicLeftShift(normal_vx_as_int, self.precision.get_exponent_size() + 1)
rcp_idx = BitLogicRightShift(rcp_shift, self.precision.get_exponent_size() + 1 + self.precision.get_field_size() - int(self.tbl_index_size))
rcp_m = TableLoad(rcp_table, rcp_idx, tag = 'rcp_idx',
debug = debug_multi)
#
rcp_m.set_attributes(tag = 'rcp_m')
# exponent is normally either 0 or -1, since m is in [1, 2). Possible
# optimization?
# exponent = ExponentExtraction(rcp_m, precision = self.precision,
# tag = 'exponent')
ri_round = TypeCast(
Addition(
TypeCast(rcp_m, precision = int_prec),
table_mantissa_half_ulp,
precision = int_prec
),
precision = uint_prec
)
ri_fast_rndn = BitLogicAnd(
ri_round,
table_s_exp_index_mask,
tag = 'ri_fast_rndn',
precision = uint_prec
)
# u = m * ri - 1
ul = None
if self.no_rcp == True: # ... u does not fit on a single word
tmp_u, tmp_ul = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = (self.no_fma == False))
fp_minus_one = Constant(-1.0, precision = self.precision, tag = 'fp_minus_one')
u, ul = Add212(fp_minus_one, tmp_u, tmp_ul)
u.set_attributes(tag='uh')
ul.set_attributes(tag='ul')
elif self.no_fma == False:
u = FusedMultiplyAdd(
vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fp_one,
specifier = FusedMultiplyAdd.Subtract,
tag = 'u')
else: # disable FMA
# tmph + tmpl = m * ri, where tmph ~ 1
tmph, tmpl = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = False)
# u_tmp = tmph - 1 ... exact due to Sterbenz
u_tmp = Subtraction(tmph, fp_one, precision = self.precision)
# u = u_tmp - tmpl ... exact since the result u is representable as a single word
u = Addition(u_tmp, tmpl, precision = self.precision, tag = 'u')
unneeded_bits = Constant(
self.precision.field_size - table_index_size,
precision=uint_prec,
tag="unneeded_bits"
)
assert self.precision.field_size - table_index_size >= 0
ri_bits = BitLogicRightShift(
ri_fast_rndn,
unneeded_bits,
precision = uint_prec,
tag = "ri_bits"
)
# Retrieve mantissa's MSBs + first bit of exponent, for tau computation in case
# exponent is 0 (i.e. biased 127, i.e. first bit of exponent is set.).
# In this particular case, i = 0 but tau is 1
# table_index does not need to be as long as uint_prec might be,
# try and keep it the size of size_t.
size_t_prec = ML_UInt32
signed_size_t_prec = ML_Int32
table_index_mask = Constant(
(1 << (table_index_size + 1)) - 1,
precision = size_t_prec
)
table_index = BitLogicAnd(
Conversion(ri_bits, precision = size_t_prec),
table_index_mask,
tag = 'table_index',
precision = size_t_prec
)
# Compute tau using the tau_index_limit value.
tmp = default_bool_convert(
Comparison(
TypeCast(table_index, precision = signed_size_t_prec),
Constant(tau_index_limit, precision = signed_size_t_prec),
specifier = Comparison.Greater
if isinstance(self.processor, VectorBackend)
else Comparison.LessOrEqual
),
precision = signed_size_t_prec,
tag="tmp"
)
# A true tmp will typically be -1 for VectorBackends, but 1 for standard C.
tau = Conversion(
Addition(tmp, Constant(1, precision=signed_size_t_prec), precision = signed_size_t_prec, tag="pre_add")
if isinstance(self.processor, VectorBackend)
else tmp,
precision=int_prec,
tag="pre_tau"
)
tau.set_attributes(tag = 'tau')
# Update table_index: keep only table_index_size bits
table_index_hi = BitLogicAnd(
table_index,
Constant((1 << table_index_size) - 1, precision = size_t_prec),
precision = size_t_prec
)
# table_index_hi = table_index_hi << 1
table_index_hi = BitLogicLeftShift(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_hi"
)
# table_index_lo = table_index_hi + 1
table_index_lo = Addition(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_lo"
)
tbl_hi = TableLoad(log1p_table, table_index_hi, tag = 'tbl_hi',
debug = debug_multi)
tbl_lo = TableLoad(log1p_table, table_index_lo, tag = 'tbl_lo',
debug = debug_multi)
# Compute exponent e + tau - alpha, but first subtract the bias.
if self.no_subnormal == False:
tmp_eptau = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
exponent = Subtraction(tmp_eptau, alpha, precision = int_prec)
else:
exponent = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
#
fp_exponent = Conversion(exponent, precision = self.precision,
tag = 'fp_exponent')
Log.report(Log.Info, 'MDL polynomial approximation')
if self.log_radix == EXP_1:
sollya_function = log(1 + sollya.x)
elif self.log_radix == 2:
sollya_function = log2(1 + sollya.x)
elif self.log_radix == 10:
sollya_function = log10(1 + sollya.x)
# ...
if self.force_division == True: # rcp accuracy is 2^(-p)
boundrcp = 2**(-self.precision.get_precision())
else:
boundrcp = 1.5 * 2**(-12) # ... see Intel intrinsics guide
if self.precision in [ML_Binary64]:
if not self.processor.is_supported_operation(rcp_m):
boundrcp = (1+boundrcp)*(1+2**(-24)) - 1
else:
boundrcp = 2**(-14) # ... see Intel intrinsics guide
arg_red_mag = boundrcp + 2**(-table_index_size-1) + boundrcp * 2**(-table_index_size-1)
if self.no_rcp == False:
approx_interval = Interval(-arg_red_mag, arg_red_mag)
else:
approx_interval = Interval(-2**(-int(self.tbl_index_size)+1),2**(-int(self.tbl_index_size)+1))
max_eps = 2**-(2*(self.precision.get_field_size()))
Log.report(Log.Info, "max acceptable error for polynomial = {}".format(float.hex(max_eps)))
poly_degree = sup(
guessdegree(
sollya_function,
approx_interval,
max_eps,
)
)
Log.report(Log.Info, "poly degree is ", poly_degree)
if self.log_radix == EXP_1:
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(2, int(poly_degree) + 1), # Force 1st 2 coeffs to 0 and 1, resp.
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree - 1),
approx_interval,
sollya.absolute,
0 + sollya._x_) # Force the first 2 coefficients to 0 and 1, resp.
else: # ... == '2' or '10'
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(1, int(poly_degree) + 1), # Force 1st coeff to 0
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree),
approx_interval,
sollya.absolute,
0) # Force the first coefficients to 0
Log.report(Log.Info, str(poly_object))
constant_precision = ML_SingleSingle if self.precision == ML_Binary32 \
else ML_DoubleDouble if self.precision == ML_Binary64 \
else None
if is_cgpe_available():
log1pu_poly = PolynomialSchemeEvaluator.generate_cgpe_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision, scheme_id = cgpe_scheme_idx
)
else:
Log.report(Log.Warning,
"CGPE not available, falling back to std poly evaluator")
log1pu_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision
)
# XXX Dirty implementation of double-(self.precision) poly
def dirty_poly_node_conversion(node, variable_h, variable_l, use_fma):
return dirty_multi_node_expand(
node, self.precision, mem_map={variable_h: (variable_h, variable_l)}, fma=use_fma)
log1pu_poly_hi, log1pu_poly_lo = dirty_poly_node_conversion(log1pu_poly, u, ul,
use_fma=(self.no_fma == False))
log1pu_poly_hi.set_attributes(tag = 'log1pu_poly_hi')
log1pu_poly_lo.set_attributes(tag = 'log1pu_poly_lo')
# Compute log(2) * (e + tau - alpha)
if self.log_radix != 2: # 'e' or '10'
log2e_hi, log2e_lo = Mul212(fp_exponent, log2_hi, log2_lo,
fma = (self.no_fma == False))
# Add log1p(u)
if self.log_radix != 2: # 'e' or '10'
tmp_res_hi, tmp_res_lo = Add222(log2e_hi, log2e_lo,
log1pu_poly_hi, log1pu_poly_lo)
else:
tmp_res_hi, tmp_res_lo = Add212(fp_exponent,
log1pu_poly_hi, log1pu_poly_lo)
# Add -log(2^(tau)/m) approximation retrieved by two table lookups
logx_hi = Add122(tmp_res_hi, tmp_res_lo, tbl_hi, tbl_lo)[0]
logx_hi.set_attributes(tag = 'logx_hi')
scheme = Return(logx_hi, precision = self.precision)
return scheme
def 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
def generate_scheme(self):
def compute_reciprocal(vx):
inv_seed = ReciprocalSeed(vx, precision = self.precision, tag = "inv_seed", debug = debug_multi)
nr_1 = 2*inv_seed - vx*inv_seed*inv_seed
nr_2 = 2*nr_1 - vx*nr_1*nr_1
nr_3 =2*nr_2 - vx*nr_2*nr_2
inv_vx = 2*nr_3 - vx*nr_3*nr_3
return inv_vx
vx = self.implementation.add_input_variable("x", self.get_input_precision())
sollya_precision = self.precision.get_sollya_object()
int_precision = {
ML_Binary32 : ML_Int32,
ML_Binary64 : ML_Int64
}[self.precision]
hi_precision = self.precision.get_field_size() - 12
half_pi = round(pi/2, sollya_precision, sollya.RN)
half_pi_cst = Constant(half_pi, precision = self.precision)
test_sign = Comparison(vx, 0, specifier = Comparison.Less, precision = ML_Bool, debug = debug_multi, tag = "Is_Negative")
neg_vx = -vx
sign = Variable("sign", precision = self.precision, var_type = Variable.Local)
abs_vx_std = Variable("abs_vx", precision = self.precision, var_type = Variable.Local)
red_vx_std = Variable("red_vx", precision = self.precision, var_type = Variable.Local)
const_index_std = Variable("const_index", precision = int_precision, var_type = Variable.Local)
set_sign = Statement(
ConditionBlock(test_sign,
Statement(ReferenceAssign(abs_vx_std, neg_vx), ReferenceAssign(sign, -1)),
Statement(ReferenceAssign(abs_vx_std, vx), ReferenceAssign(sign, 1))
))
if self.precision is ML_Binary32:
bound = 24
else:
bound = 53
test_bound = Comparison(abs_vx_std, S2**bound, specifier = Comparison.GreaterOrEqual, precision = ML_Bool)#, debug = debug_multi, tag ="bound")
test_bound1 = Comparison(abs_vx_std, 39.0/16.0, specifier = Comparison.GreaterOrEqual, precision = ML_Bool)#, debug = debug_multi, tag ="bound")
test_bound2 = Comparison(abs_vx_std, 19.0/16.0, specifier = Comparison.GreaterOrEqual, precision = ML_Bool)#, debug = debug_multi, tag ="bound")
test_bound3 = Comparison(abs_vx_std, 11.0/16.0, specifier = Comparison.GreaterOrEqual, precision = ML_Bool)#, debug = debug_multi, tag ="bound")
test_bound4 = Comparison(abs_vx_std, 7.0/16.0, specifier = Comparison.GreaterOrEqual, precision = ML_Bool)#, debug = debug_multi, tag ="bound")
set_bound = Return(sign*half_pi_cst)
set_bound1 = Statement(
ReferenceAssign(red_vx_std, -compute_reciprocal(abs_vx_std)),
ReferenceAssign(const_index_std, 3)
)
set_bound2 = Statement(
ReferenceAssign(red_vx_std, (abs_vx_std - 1.5)*compute_reciprocal(1 + 1.5*abs_vx_std)),
ReferenceAssign(const_index_std, 2)
)
set_bound3 = Statement(
ReferenceAssign(red_vx_std, (abs_vx_std - 1.0)*compute_reciprocal(abs_vx_std + 1.0)),
ReferenceAssign(const_index_std, 1)
)
set_bound4 = Statement(
ReferenceAssign(red_vx_std, (abs_vx_std - 0.5)*compute_reciprocal(1 + abs_vx_std*0.5)),
ReferenceAssign(const_index_std, 0)
)
set_bound5 = Statement(
ReferenceAssign(red_vx_std, abs_vx_std),
ReferenceAssign(const_index_std, 4)
)
cons_table = ML_NewTable(dimensions = [5, 2], storage_precision = self.precision, tag = self.uniquify_name("cons_table"))
coeff_table = ML_NewTable(dimensions = [11], storage_precision = self.precision, tag = self.uniquify_name("coeff_table"))
cons_hi = round(atan(0.5), hi_precision, sollya.RN)
cons_table[0][0] = cons_hi
cons_table[0][1] = round(atan(0.5) - cons_hi, sollya_precision, sollya.RN)
cons_hi = round(atan(1.0), hi_precision, sollya.RN)
cons_table[1][0] = cons_hi
cons_table[1][1] = round(atan(1.0) - cons_hi, sollya_precision, sollya.RN)
cons_hi = round(atan(1.5), hi_precision, sollya.RN)
cons_table[2][0] = cons_hi
cons_table[2][1] = round(atan(1.5) - cons_hi, sollya_precision, sollya.RN)
cons_hi = round(pi/2, hi_precision, sollya.RN)
cons_table[3][0] = cons_hi
cons_table[3][1] = round(pi/2 - cons_hi, sollya_precision, sollya.RN)
cons_table[4][0] = 0.0
cons_table[4][1] = 0.0
coeff_table[0] = round(3.33333333333329318027e-01, sollya_precision, sollya.RN)
coeff_table[1] = round(-1.99999999998764832476e-01, sollya_precision, sollya.RN)
coeff_table[2] = round(1.42857142725034663711e-01, sollya_precision, sollya.RN)
coeff_table[3] = round(-1.11111104054623557880e-01, sollya_precision, sollya.RN)
coeff_table[4] = round(9.09088713343650656196e-02, sollya_precision, sollya.RN)
coeff_table[5] = round(-7.69187620504482999495e-02, sollya_precision, sollya.RN)
coeff_table[6] = round(6.66107313738753120669e-02, sollya_precision, sollya.RN)
coeff_table[7] = round(-5.83357013379057348645e-02, sollya_precision, sollya.RN)
coeff_table[8] = round(4.97687799461593236017e-02, sollya_precision, sollya.RN)
coeff_table[9] = round(-3.65315727442169155270e-02, sollya_precision, sollya.RN)
coeff_table[10] = round(1.62858201153657823623e-02, sollya_precision, sollya.RN)
red_vx2 = red_vx_std*red_vx_std
red_vx4 = red_vx2*red_vx2
a0 = TableLoad(coeff_table, 0, precision = self.precision)
a1 = TableLoad(coeff_table, 1, precision = self.precision)
a2 = TableLoad(coeff_table, 2, precision = self.precision)
a3 = TableLoad(coeff_table, 3, precision = self.precision)
a4 = TableLoad(coeff_table, 4, precision = self.precision)
a5 = TableLoad(coeff_table, 5, precision = self.precision)
a6 = TableLoad(coeff_table, 6, precision = self.precision)
a7 = TableLoad(coeff_table, 7, precision = self.precision)
a8 = TableLoad(coeff_table, 8, precision = self.precision)
a9 = TableLoad(coeff_table, 9, precision = self.precision)
a10 = TableLoad(coeff_table, 10, precision = self.precision)
poly_even = red_vx2*(a0 + red_vx4*(a2 + red_vx4*(a4 + red_vx4*(a6 + red_vx4*(a8 + red_vx4*a10)))))
poly_odd = red_vx4*(a1 + red_vx4*(a3 + red_vx4*(a5 + red_vx4*(a7 + red_vx4*a9))))
poly_even.set_attributes(tag = "poly_even", debug = debug_multi)
poly_odd.set_attributes(tag = "poly_odd", debug = debug_multi)
const_load_hi = TableLoad(cons_table, const_index_std, 0, tag = "const_load_hi", debug = debug_multi)
const_load_lo = TableLoad(cons_table, const_index_std, 1, tag = "const_load_lo", debug = debug_multi)
test_NaN_or_inf = Test(vx, specifier = Test.IsInfOrNaN, tag = "nan_or_inf", likely = False)
test_nan = Test(vx, specifier = Test.IsNaN, debug = debug_multi, tag = "is_nan_test", likely = False)
test_positive = Comparison(vx, 0, specifier = Comparison.GreaterOrEqual, debug = debug_multi, tag = "inf_sign", likely = False)
result = const_load_hi - ((red_vx_std*(poly_even + poly_odd) - const_load_lo) - red_vx_std)
result.set_attributes(tag = "result", debug = debug_multi)
std_scheme = Statement(
sign,
abs_vx_std,
red_vx_std,
const_index_std,
set_sign,
ConditionBlock(
test_bound,
set_bound,
ConditionBlock(
test_bound1,
set_bound1,
ConditionBlock(
test_bound2,
set_bound2,
ConditionBlock(
test_bound3,
set_bound3,
ConditionBlock(
test_bound4,
set_bound4,
set_bound5
)
)
)
)
),
Return(sign*result)
)
infty_return = ConditionBlock(test_positive, Return(half_pi_cst), Return(-half_pi_cst))
non_std_return = ConditionBlock(test_nan, Return(FP_QNaN(self.precision)), infty_return)
scheme = ConditionBlock(test_NaN_or_inf, Statement(ClearException(), non_std_return), std_scheme)
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_scheme(self):
# declaring CodeFunction and retrieving input variable
vx = self.implementation.add_input_variable("x", self.precision)
Log.report(Log.Info, "target: %s " % self.processor.target_name)
# display parameter information
Log.report(Log.Info, "accuracy : %s " % self.accuracy)
Log.report(Log.Info, "input interval: %s " % self.input_interval)
accuracy_goal = self.accuracy.get_goal()
Log.report(Log.Info, "accuracy_goal=%f" % accuracy_goal)
table_size_log = self.table_size_log
integer_size = 31
integer_precision = ML_Int32
max_bound = sup(abs(self.input_interval))
max_bound_log = int(ceil(log2(max_bound)))
Log.report(Log.Info, "max_bound_log=%s " % max_bound_log)
scaling_power = integer_size - max_bound_log
Log.report(Log.Info, "scaling power: %s " % scaling_power)
storage_precision = ML_Custom_FixedPoint_Format(1, 30, signed=True)
Log.report(Log.Info, "tabulating cosine and sine")
# cosine and sine fused table
fused_table = ML_NewTable(
dimensions=[2**table_size_log, 2],
storage_precision=storage_precision,
tag="fast_lib_shared_table") # self.uniquify_name("cossin_table"))
# filling table
for i in range(2**table_size_log):
local_x = i / S2**table_size_log * S2**max_bound_log
cos_local = cos(
local_x
) # nearestint(cos(local_x) * S2**storage_precision.get_frac_size())
sin_local = sin(
local_x
) # nearestint(sin(local_x) * S2**storage_precision.get_frac_size())
fused_table[i][0] = cos_local
fused_table[i][1] = sin_local
# argument reduction evaluation scheme
# scaling_factor = Constant(S2**scaling_power, precision = self.precision)
red_vx_precision = ML_Custom_FixedPoint_Format(31 - scaling_power,
scaling_power,
signed=True)
Log.report(
Log.Verbose, "red_vx_precision.get_c_bit_size()=%d" %
red_vx_precision.get_c_bit_size())
# red_vx = NearestInteger(vx * scaling_factor, precision = integer_precision)
red_vx = Conversion(vx,
precision=red_vx_precision,
tag="red_vx",
debug=debug_fixed32)
computation_precision = red_vx_precision # self.precision
output_precision = self.io_precisions[0]
Log.report(Log.Info,
"computation_precision is %s" % computation_precision)
Log.report(Log.Info, "storage_precision is %s" % storage_precision)
Log.report(Log.Info, "output_precision is %s" % output_precision)
hi_mask_value = 2**32 - 2**(32 - table_size_log - 1)
hi_mask = Constant(hi_mask_value, precision=ML_Int32)
Log.report(Log.Info, "hi_mask=0x%x" % hi_mask_value)
red_vx_hi_int = BitLogicAnd(TypeCast(red_vx, precision=ML_Int32),
hi_mask,
precision=ML_Int32,
tag="red_vx_hi_int",
debug=debugd)
red_vx_hi = TypeCast(red_vx_hi_int,
precision=red_vx_precision,
tag="red_vx_hi",
debug=debug_fixed32)
red_vx_lo = red_vx - red_vx_hi
red_vx_lo.set_attributes(precision=red_vx_precision,
tag="red_vx_lo",
debug=debug_fixed32)
table_index = BitLogicRightShift(TypeCast(red_vx, precision=ML_Int32),
scaling_power -
(table_size_log - max_bound_log),
precision=ML_Int32,
tag="table_index",
debug=debugd)
tabulated_cos = TableLoad(fused_table,
table_index,
0,
tag="tab_cos",
precision=storage_precision,
debug=debug_fixed32)
tabulated_sin = TableLoad(fused_table,
table_index,
1,
tag="tab_sin",
precision=storage_precision,
debug=debug_fixed32)
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
Log.report(Log.Info, "building polynomial approximation for cosine")
# cosine polynomial approximation
poly_interval = Interval(0, S2**(max_bound_log - table_size_log))
Log.report(Log.Info, "poly_interval=%s " % poly_interval)
cos_poly_degree = 2 # int(sup(guessdegree(cos(x), poly_interval, accuracy_goal)))
Log.report(Log.Verbose, "cosine polynomial approximation")
cos_poly_object, cos_approx_error = Polynomial.build_from_approximation_with_error(
cos(x), [0, 2], [0] + [computation_precision.get_bit_size()],
poly_interval,
sollya.absolute,
error_function=error_function)
#cos_eval_scheme = PolynomialSchemeEvaluator.generate_horner_scheme(cos_poly_object, red_vx_lo, unified_precision = computation_precision)
Log.report(Log.Info, "cos_approx_error=%e" % cos_approx_error)
cos_coeff_list = cos_poly_object.get_ordered_coeff_list()
coeff_C0 = cos_coeff_list[0][1]
coeff_C2 = Constant(cos_coeff_list[1][1],
precision=ML_Custom_FixedPoint_Format(-1,
32,
signed=True))
Log.report(Log.Info, "building polynomial approximation for sine")
# sine polynomial approximation
sin_poly_degree = 2 # int(sup(guessdegree(sin(x)/x, poly_interval, accuracy_goal)))
Log.report(Log.Info, "sine poly degree: %e" % sin_poly_degree)
Log.report(Log.Verbose, "sine polynomial approximation")
sin_poly_object, sin_approx_error = Polynomial.build_from_approximation_with_error(
sin(sollya.x) / sollya.x, [0, 2], [0] +
[computation_precision.get_bit_size()] * (sin_poly_degree + 1),
poly_interval,
sollya.absolute,
error_function=error_function)
sin_coeff_list = sin_poly_object.get_ordered_coeff_list()
coeff_S0 = sin_coeff_list[0][1]
coeff_S2 = Constant(sin_coeff_list[1][1],
precision=ML_Custom_FixedPoint_Format(-1,
32,
signed=True))
# scheme selection between sine and cosine
if self.cos_output:
scheme = self.generate_cos_scheme(computation_precision,
tabulated_cos, tabulated_sin,
coeff_S2, coeff_C2, red_vx_lo)
else:
scheme = self.generate_sin_scheme(computation_precision,
tabulated_cos, tabulated_sin,
coeff_S2, coeff_C2, red_vx_lo)
result = Conversion(scheme, precision=self.io_precisions[0])
Log.report(
Log.Verbose, "result operation tree :\n %s " % result.get_str(
display_precision=True, depth=None, memoization_map={}))
scheme = Statement(Return(result))
return scheme
def generate_scheme(self):
# declaring target and instantiating optimization engine
vx = self.implementation.add_input_variable("x", self.precision)
Log.set_dump_stdout(True)
Log.report(Log.Info,
"\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
index_size = 3
approx_interval = Interval(0.0, 2**-index_size)
error_goal_approx = 2**-(self.precision.get_precision())
int_precision = self.precision.get_integer_format()
vx_int = Floor(vx * 2**index_size,
precision=self.precision,
tag="vx_int",
debug=debug_multi)
vx_frac = vx - (vx_int * 2**-index_size)
vx_frac.set_attributes(tag="vx_frac",
debug=debug_multi,
unbreakable=True)
poly_degree = sup(
guessdegree(2**(sollya.x), approx_interval, error_goal_approx)) + 1
precision_list = [1] + [self.precision] * (poly_degree)
vx_integer = Conversion(vx_int,
precision=int_precision,
tag="vx_integer",
debug=debug_multi)
vx_int_hi = BitLogicRightShift(vx_integer,
Constant(index_size),
tag="vx_int_hi",
debug=debug_multi)
vx_int_lo = Modulo(vx_integer,
2**index_size,
tag="vx_int_lo",
debug=debug_multi)
pow_exp = ExponentInsertion(Conversion(vx_int_hi,
precision=int_precision),
precision=self.precision,
tag="pow_exp",
debug=debug_multi)
exp2_table = ML_NewTable(dimensions=[2 * 2**index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("exp2_table"))
for i in range(2 * 2**index_size):
input_value = i - 2**index_size if i >= 2**index_size else i
exp2_value = SollyaObject(2)**((input_value) * 2**-index_size)
hi_value = round(exp2_value, self.precision.get_sollya_object(),
RN)
lo_value = round(exp2_value - hi_value,
self.precision.get_sollya_object(), RN)
exp2_table[i][0] = lo_value
exp2_table[i][1] = hi_value
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(
2**(sollya.x),
poly_degree,
precision_list,
approx_interval,
sollya.absolute,
error_function=error_function)
print "poly_approx_error: ", poly_approx_error, float(
log2(poly_approx_error))
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly = polynomial_scheme_builder(poly_object.sub_poly(start_index=1),
vx_frac,
unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
table_index = Addition(vx_int_lo,
Constant(2**index_size,
precision=int_precision),
precision=int_precision,
tag="table_index",
debug=debug_multi)
lo_value_load = TableLoad(exp2_table,
table_index,
0,
tag="lo_value_load",
debug=debug_multi)
hi_value_load = TableLoad(exp2_table,
table_index,
1,
tag="hi_value_load",
debug=debug_multi)
result = (hi_value_load +
(hi_value_load * poly +
(lo_value_load + lo_value_load * poly))) * pow_exp
ov_flag = Comparison(vx_int_hi,
Constant(self.precision.get_emax(),
precision=self.precision),
specifier=Comparison.Greater)
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = Statement(
Return(Select(ov_flag, FP_PlusInfty(self.precision), result)))
return scheme
def generate_scheme(self):
def get_virtual_cst(prec, value, language):
return prec.get_support_format().get_cst(
prec.get_base_format().get_integer_coding(value, language))
## convert @p value from an input floating-point precision
# @p in_precision to an output support format @p out_precision
io_precision = VirtualFormat(base_format=self.precision,
support_format=ML_StdLogicVectorFormat(
self.precision.get_bit_size()),
get_cst=get_virtual_cst)
# declaring main input variable
vx = self.implementation.add_input_signal("x", io_precision)
# rounding mode input
rnd_mode = self.implementation.add_input_signal(
"rnd_mode", rnd_mode_format)
if self.pipelined:
self.implementation.add_input_signal("reset", ML_StdLogic)
vx_precision = self.precision
p = vx_precision.get_mantissa_size()
exp_size = vx_precision.get_exponent_size()
exp_vx_precision = ML_StdLogicVectorFormat(
vx_precision.get_exponent_size())
mant_vx_precision = ML_StdLogicVectorFormat(p)
# fixed-point precision for operand's exponent
exp_fixed_precision = fixed_point(exp_size, 0, signed=False)
# mantissa extraction
mant_vx = TypeCast(MantissaExtraction(vx,
precision=mant_vx_precision,
tag="extracted_mantissa"),
precision=fixed_point(1, p - 1, signed=False),
debug=debug_fixed,
tag="mant_vx")
# exponent extraction
exp_vx = TypeCast(RawExponentExtraction(vx,
precision=exp_vx_precision,
tag="exp_vx"),
precision=exp_fixed_precision)
approx_index_size = 8
approx_precision = fixed_point(
2,
approx_index_size,
)
# selecting table index from input mantissa MSBs
tab_index = SubSignalSelection(mant_vx,
p - 2 - approx_index_size + 1,
p - 2,
tag="tab_index")
# declaring reciprocal approximation table
inv_approx_table = ML_NewTable(dimensions=[2**approx_index_size],
storage_precision=approx_precision,
tag="inv_approx_table")
for i in range(2**approx_index_size):
num_input = 1 + i * S2**-approx_index_size
table_value = io_precision.get_base_format().round_sollya_object(
1 / num_input)
inv_approx_table[i] = table_value
# extracting initial reciprocal approximation
inv_approx_value = TableLoad(inv_approx_table,
tab_index,
precision=approx_precision,
tag="inv_approx_value",
debug=debug_fixed)
#inv_approx_value = TypeCast(inv_approx_value, precision = approx_precision)
pre_it0_input = zext(
SubSignalSelection(mant_vx,
p - 1 - approx_index_size,
p - 1,
tag="it0_input"), 1)
it0_input = TypeCast(pre_it0_input,
precision=approx_precision,
tag="it0_input",
debug=debug_fixed)
it1_precision = RTL_FixedPointFormat(
2,
2 * approx_index_size,
support_format=ML_StdLogicVectorFormat(2 + 2 * approx_index_size))
it1_input = mant_vx
final_approx = generate_NR_iteration(
mant_vx,
inv_approx_value,
(2, approx_index_size * 2), # mult precision
(-3, 2 * approx_index_size), # error precision
(2, approx_index_size * 3), # new-approx mult
(2, approx_index_size * 2), # new approx precision
self.implementation,
pipelined=0, #1 if self.pipelined else 0,
tag_suffix="_first")
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx = generate_NR_iteration(
mant_vx,
final_approx,
# mult precision
(2, approx_index_size * 3),
# error precision
(-6, approx_index_size * 3),
# approx mult precision
(2, approx_index_size * 3),
# new approx precision
(2, approx_index_size * 3),
self.implementation,
pipelined=1 if self.pipelined else 0,
tag_suffix="_second")
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx = generate_NR_iteration(
mant_vx,
final_approx,
# mult-precision
(2, 2 * p - 1),
# error precision
(-(3 * approx_index_size) / 2, approx_index_size * 2 + p - 1),
# mult approx mult precision
(2, approx_index_size * 2 + p - 1),
# approx precision
(2, p),
self.implementation,
pipelined=2 if self.pipelined else 0,
tag_suffix="_third")
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx = generate_NR_iteration(
mant_vx,
final_approx, (2, 2 * p), (-(4 * p) / 5, 2 * p), (2, 2 * p),
(2, 2 * p),
self.implementation,
pipelined=2 if self.pipelined else 0,
tag_suffix="_last")
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx.set_attributes(tag="final_approx", debug=debug_hex)
last_approx_norm = final_approx
offset_bit = BitSelection(last_approx_norm,
FixedPointPosition(
last_approx_norm,
0,
align=FixedPointPosition.FromPointToLSB),
tag="offset_bit",
debug=debug_std)
# extracting bit to determine if result should be left-shifted and
# exponent incremented
not_decrement = offset_bit
final_approx_reduced = SubSignalSelection(
final_approx,
FixedPointPosition(final_approx,
-(p - 1),
align=FixedPointPosition.FromPointToLSB),
FixedPointPosition(final_approx,
0,
align=FixedPointPosition.FromPointToLSB),
precision=fixed_point(p, 0, signed=False))
final_approx_reduced_shifted = SubSignalSelection(
final_approx,
FixedPointPosition(final_approx,
-p,
align=FixedPointPosition.FromPointToLSB),
FixedPointPosition(final_approx,
-1,
align=FixedPointPosition.FromPointToLSB),
precision=fixed_point(p, 0, signed=False))
# unrounded mantissa field excluding leading digit
unrounded_mant_field = Select(
equal_to(not_decrement, 1),
final_approx_reduced,
final_approx_reduced_shifted,
precision=fixed_point(p, 0, signed=False),
tag="unrounded_mant_field",
debug=debug_hex,
)
def get_bit(optree, bit_index):
bit_sel = BitSelection(
optree,
FixedPointPosition(optree,
-bit_index,
align=FixedPointPosition.FromPointToLSB))
return bit_sel
mant_lsb = Select(
equal_to(not_decrement, 1),
get_bit(final_approx, p - 1),
get_bit(final_approx, p),
precision=ML_StdLogic,
tag="mant_lsb",
debug=debug_std,
)
round_bit = Select(
equal_to(not_decrement, 1),
get_bit(final_approx, p),
get_bit(final_approx, p + 1),
precision=ML_StdLogic,
tag="round_bit",
debug=debug_std,
)
sticky_bit_input = Select(
equal_to(not_decrement, 1),
SubSignalSelection(final_approx,
0,
FixedPointPosition(
final_approx,
-(p + 1),
align=FixedPointPosition.FromPointToLSB),
precision=None,
tag="sticky_bit_input"),
SubSignalSelection(final_approx,
0,
FixedPointPosition(
final_approx,
-(p + 2),
align=FixedPointPosition.FromPointToLSB),
precision=None,
tag="sticky_bit_input"),
)
sticky_bit = Select(Equal(sticky_bit_input, Constant(0,
precision=None)),
Constant(0, precision=ML_StdLogic),
Constant(1, precision=ML_StdLogic),
precision=ML_StdLogic,
tag="sticky_bit",
debug=debug_std)
# TODO: manage leading digit (in case of subnormal result)
pre_result = unrounded_mant_field
# real_exp = exp_vx - bias
# - real_exp = bias - exp_vx
# encoded negated exp = bias - exp_vx + bias = 2 * bias - exp_vx
fp_io_precision = io_precision.get_base_format()
neg_exp = -2 * fp_io_precision.get_bias() - exp_vx
neg_exp.set_attributes(tag="neg_exp", debug=debug_fixed)
res_exp = Subtraction(neg_exp,
Select(equal_to(not_decrement, 1),
Constant(0,
precision=exp_fixed_precision),
Constant(1,
precision=exp_fixed_precision),
precision=None,
tag="exp_offset",
debug=debug_fixed),
tag="res_exp",
debug=debug_fixed)
res_exp_field = SubSignalSelection(
res_exp,
FixedPointPosition(res_exp,
0,
align=FixedPointPosition.FromPointToLSB,
tag="res_exp_field LSB"),
FixedPointPosition(res_exp,
exp_size - 1,
align=FixedPointPosition.FromPointToLSB,
tag="res_exp_field MSB"),
precision=None,
tag="res_exp_field",
# debug=debug_fixed
)
result_sign = CopySign(vx, precision=ML_StdLogic)
exp_mant_precision = ML_StdLogicVectorFormat(
io_precision.get_bit_size() - 1)
rnd_mode_is_rne = Equal(rnd_mode, rnd_rne, precision=ML_Bool)
rnd_mode_is_ru = Equal(rnd_mode, rnd_ru, precision=ML_Bool)
rnd_mode_is_rd = Equal(rnd_mode, rnd_rd, precision=ML_Bool)
rnd_mode_is_rz = Equal(rnd_mode, rnd_rz, precision=ML_Bool)
round_incr = Conversion(
logical_or_reduce([
logical_and_reduce([
rnd_mode_is_rne,
equal_to(round_bit, 1),
equal_to(sticky_bit, 1)
]),
logical_and_reduce([
rnd_mode_is_rne,
equal_to(round_bit, 1),
equal_to(sticky_bit, 0),
equal_to(mant_lsb, 1)
]),
logical_and_reduce([
rnd_mode_is_ru,
equal_to(result_sign, 0),
LogicalOr(equal_to(round_bit, 1),
equal_to(sticky_bit, 1),
precision=ML_Bool)
]),
logical_and_reduce([
rnd_mode_is_rd,
equal_to(result_sign, 1),
LogicalOr(equal_to(round_bit, 1),
equal_to(sticky_bit, 1),
precision=ML_Bool)
]),
]),
precision=fixed_point(1, 0, signed=False),
tag="round_incr",
#debug=debug_fixed
)
# Precision for result without sign
unsigned_result_prec = fixed_point((p - 1) + exp_size, 0)
unrounded_mant_field_nomsb = Conversion(
unrounded_mant_field,
precision=fixed_point(p - 1, 0, signed=False),
tag="unrounded_mant_field_nomsb",
debug=debug_hex)
pre_rounded_unsigned_result = Concatenation(
res_exp_field,
unrounded_mant_field_nomsb,
precision=unsigned_result_prec,
tag="pre_rounded_unsigned_result")
unsigned_result_rounded = Addition(pre_rounded_unsigned_result,
round_incr,
precision=unsigned_result_prec,
tag="unsigned_result")
vr_out = TypeCast(Concatenation(
result_sign,
TypeCast(unsigned_result_rounded,
precision=ML_StdLogicVectorFormat(p - 1 + exp_size)),
precision=ML_StdLogicVectorFormat(io_precision.get_bit_size())),
precision=io_precision,
debug=debug_hex,
tag="vr_out")
self.implementation.add_output_signal("vr_out", vr_out)
return [self.implementation]
def generate_scheme(self):
def get_virtual_cst(prec, value, language):
return prec.get_support_format().get_cst(
prec.get_base_format().get_integer_coding(value, language))
## convert @p value from an input floating-point precision
# @p in_precision to an output support format @p out_precision
io_precision = HdlVirtualFormat(self.precision)
# declaring main input variable
vx = self.implementation.add_input_signal("x", io_precision)
if self.pipelined:
self.implementation.add_input_signal("reset", ML_StdLogic)
vx_precision = self.precision
p = vx_precision.get_mantissa_size()
exp_vx_precision = ML_StdLogicVectorFormat(vx_precision.get_exponent_size())
mant_vx_precision = ML_StdLogicVectorFormat(p)
# mantissa extraction
mant_vx = MantissaExtraction(vx, precision = mant_vx_precision, tag = "mant_vx")
# exponent extraction
exp_vx = RawExponentExtraction(vx, precision = exp_vx_precision, tag = "exp_vx", debug = debug_dec)
approx_index_size = 8
approx_precision = RTL_FixedPointFormat(
2, approx_index_size,
support_format = ML_StdLogicVectorFormat(approx_index_size + 2),
)
# selecting table index from input mantissa MSBs
tab_index = SubSignalSelection(mant_vx, p-2 - approx_index_size +1, p-2, tag = "tab_index")
# declaring reciprocal approximation table
inv_approx_table = ML_NewTable(dimensions = [2**approx_index_size], storage_precision = approx_precision, tag = "inv_approx_table")
for i in range(2**approx_index_size):
num_input = 1 + i * S2**-approx_index_size
table_value = io_precision.get_base_format().round_sollya_object(1 / num_input)
inv_approx_table[i] = table_value
# extracting initial reciprocal approximation
inv_approx_value = TableLoad(inv_approx_table, tab_index, precision = approx_precision, tag = "inv_approx_value", debug = debug_fixed)
#inv_approx_value = TypeCast(inv_approx_value, precision = approx_precision)
pre_it0_input = zext(SubSignalSelection(mant_vx, p-1 - approx_index_size , p-1, tag = "it0_input"), 1)
it0_input = TypeCast(pre_it0_input, precision = approx_precision, tag = "it0_input", debug = debug_fixed)
it1_precision = RTL_FixedPointFormat(
2,
2 * approx_index_size,
support_format = ML_StdLogicVectorFormat(2 + 2 * approx_index_size)
)
pre_it1_input = zext(SubSignalSelection(mant_vx, p - 1 - 2 * approx_index_size, p -1, tag = "it1_input"), 1)
it1_input = TypeCast(pre_it1_input, precision = it1_precision, tag = "it1_input", debug = debug_fixed)
final_approx = generate_NR_iteration(
it0_input,
inv_approx_value,
(2, approx_index_size * 2), # mult precision
(-3, 2 * approx_index_size), # error precision
(2, approx_index_size * 3), # new-approx mult
(2, approx_index_size * 2), # new approx precision
self.implementation,
pipelined = 0, #1 if self.pipelined else 0,
tag_suffix = "_first"
)
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx = generate_NR_iteration(
it1_input,
final_approx,
# mult precision
(2, approx_index_size * 3),
# error precision
(-6, approx_index_size * 3),
# approx mult precision
(2, approx_index_size * 3),
# new approx precision
(2, approx_index_size * 3),
self.implementation,
pipelined = 1 if self.pipelined else 0,
tag_suffix = "_second"
)
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
last_it_precision = RTL_FixedPointFormat(
2,
p - 1,
support_format=ML_StdLogicVectorFormat(2 + p - 1)
)
pre_last_it_input = zext(mant_vx, 1)
last_it_input = TypeCast(
pre_last_it_input, precision=last_it_precision,
tag="last_it_input", debug=debug_fixed
)
final_approx = generate_NR_iteration(
last_it_input,
final_approx,
# mult-precision
(2, 2 * p - 1),
# error precision
(int(- (3 * approx_index_size) / 2), approx_index_size * 2 + p - 1),
# mult approx mult precision
(2, approx_index_size * 2 + p - 1),
# approx precision
(2, p),
self.implementation,
pipelined = 2 if self.pipelined else 0,
tag_suffix = "_third"
)
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx = generate_NR_iteration(
last_it_input,
final_approx,
(2, 2 * p),
(int(-(4 * p)/5), 2 * p),
(2, 2 * p),
(2, 2 * p),
self.implementation,
pipelined = 2 if self.pipelined else 0,
tag_suffix = "_last"
)
# Inserting post-input pipeline stage
if self.pipelined: self.implementation.start_new_stage()
final_approx.set_attributes(tag = "final_approx", debug = debug_fixed)
# bit indexes to select mantissa from final_approximation
pre_mant_size = min(self.precision.get_field_size(), final_approx.get_precision().get_frac_size())
final_approx_frac_msb_index = final_approx.get_precision().get_frac_size() - 1
final_approx_frac_lsb_index = final_approx.get_precision().get_frac_size() - pre_mant_size
# extracting bit to determine if result should be left-shifted and
# exponent incremented
cst_index = Constant(final_approx.get_precision().get_frac_size(), precision = ML_Integer)
final_approx_casted = TypeCast(final_approx, precision = ML_StdLogicVectorFormat(final_approx.get_precision().get_bit_size()))
not_decrement = final_approx_casted[cst_index]
not_decrement.set_attributes(precision = ML_StdLogic, tag = "not_decrement", debug = debug_std)
logic_1 = Constant(1, precision = ML_StdLogic)
result = Select(
Comparison( not_decrement, logic_1, specifier = Comparison.Equal, precision = ML_Bool),
SubSignalSelection(
TypeCast(
final_approx,
precision = ML_StdLogicVectorFormat(final_approx.get_precision().get_bit_size())
),
final_approx_frac_lsb_index,
final_approx_frac_msb_index,
),
SubSignalSelection(
TypeCast(
final_approx,
precision = ML_StdLogicVectorFormat(final_approx.get_precision().get_bit_size())
),
final_approx_frac_lsb_index - 1,
final_approx_frac_msb_index - 1,
),
precision = ML_StdLogicVectorFormat(pre_mant_size),
tag = "result"
)
def get_bit(optree, bit_index):
bit_index_cst = Constant(bit_index, precision = ML_Integer)
bit_sel = VectorElementSelection(
optree,
bit_index_cst,
precision = ML_StdLogic)
return bit_sel
least_bit = Select(
Comparison(not_decrement, logic_1, specifier = Comparison.Equal, precision = ML_Bool),
get_bit(final_approx_casted, final_approx_frac_lsb_index),
get_bit(final_approx_casted, final_approx_frac_lsb_index - 1),
precision = ML_StdLogic,
tag = "least_bit",
debug = debug_std,
)
round_bit = Select(
Comparison(not_decrement, logic_1, specifier = Comparison.Equal, precision = ML_Bool),
get_bit(final_approx_casted, final_approx_frac_lsb_index - 1),
get_bit(final_approx_casted, final_approx_frac_lsb_index - 2),
precision = ML_StdLogic,
tag = "round_bit",
debug = debug_std,
)
sticky_bit_input = Select(
Comparison(not_decrement, logic_1, specifier = Comparison.Equal, precision = ML_Bool),
SubSignalSelection(
final_approx_casted, 0,
final_approx_frac_lsb_index - 2,
precision = ML_StdLogicVectorFormat(final_approx_frac_lsb_index - 1)
),
zext(
SubSignalSelection(
final_approx_casted, 0,
final_approx_frac_lsb_index - 3,
precision = ML_StdLogicVectorFormat(final_approx_frac_lsb_index - 2)
),
1
),
precision = ML_StdLogicVectorFormat(final_approx_frac_lsb_index - 1)
)
sticky_bit = Select(
Equal(
sticky_bit_input,
Constant(0, precision = ML_StdLogicVectorFormat(final_approx_frac_lsb_index - 1))
),
Constant(0, precision = ML_StdLogic),
Constant(1, precision = ML_StdLogic),
precision = ML_StdLogic,
tag = "sticky_bit",
debug = debug_std
)
# if mantissa require extension
if pre_mant_size < self.precision.get_mantissa_size() - 1:
result = rzext(result, self.precision.get_mantissa_size() - 1 - pre_mant_size)
res_mant_field = result
# real_exp = exp_vx - bias
# - real_exp = bias - exp_vx
# encoded negated exp = bias - exp_vx + bias = 2 * bias - exp_vx
fp_io_precision = io_precision.get_base_format()
exp_op_precision = ML_StdLogicVectorFormat(fp_io_precision.get_exponent_size() + 2)
biasX2 = Constant(- 2 * fp_io_precision.get_bias(), precision = exp_op_precision)
neg_exp = Subtraction(
SignCast(
biasX2,
specifier = SignCast.Unsigned,
precision = get_unsigned_precision(exp_op_precision)
),
SignCast(
zext(exp_vx, 2),
specifier = SignCast.Unsigned,
precision = get_unsigned_precision(exp_op_precision),
),
precision = exp_op_precision,
tag = "neg_exp",
debug = debug_dec
)
neg_exp_field = SubSignalSelection(
neg_exp,
0,
fp_io_precision.get_exponent_size() - 1,
precision = ML_StdLogicVectorFormat(fp_io_precision.get_exponent_size())
)
res_exp = Addition(
SignCast(
neg_exp_field,
precision = get_unsigned_precision(exp_vx.get_precision()),
specifier = SignCast.Unsigned
),
SignCast(
Select(
Comparison(not_decrement, logic_1, specifier = Comparison.Equal, precision = ML_Bool),
Constant(0, precision = exp_vx_precision),
Constant(-1, precision = exp_vx_precision),
precision = exp_vx_precision
),
precision = get_unsigned_precision(exp_vx_precision),
specifier = SignCast.Unsigned
),
precision = exp_vx_precision,
tag = "result_exp",
debug = debug_dec
)
res_sign = CopySign(vx, precision = ML_StdLogic)
exp_mant_precision = ML_StdLogicVectorFormat(io_precision.get_bit_size() - 1)
round_incr = Select(
LogicalAnd(
Equal(round_bit, Constant(1, precision = ML_StdLogic)),
LogicalOr(
Equal(sticky_bit, Constant(1, precision = ML_StdLogic)),
Equal(least_bit, Constant(1, precision = ML_StdLogic)),
precision = ML_Bool,
),
precision = ML_Bool,
),
Constant(1, precision = ML_StdLogic),
Constant(0, precision = ML_StdLogic),
tag = "round_incr",
precision = ML_StdLogic,
debug = debug_std
)
exp_mant = Concatenation(
res_exp,
res_mant_field,
precision = exp_mant_precision
)
exp_mant_rounded = Addition(
SignCast(
exp_mant,
SignCast.Unsigned,
precision = get_unsigned_precision(exp_mant_precision)
),
round_incr,
precision = exp_mant_precision,
tag = "exp_mant_rounded"
)
vr_out = TypeCast(
Concatenation(
res_sign,
exp_mant_rounded,
precision = ML_StdLogicVectorFormat(io_precision.get_bit_size())
),
precision = io_precision,
debug = debug_hex,
tag = "vr_out"
)
self.implementation.add_output_signal("vr_out", vr_out)
return [self.implementation]
def generate_scalar_scheme(self, vx):
Log.set_dump_stdout(True)
Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m")
if self.debug_flag:
Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m")
index_size = 5
comp_lo = (vx < 0)
comp_lo.set_attributes(tag = "comp_lo", precision = ML_Bool)
sign = Select(comp_lo, -1, 1, precision = self.precision)
# as sinh is an odd function, we can simplify the input to its absolute
# value once the sign has been extracted
vx = Abs(vx)
int_precision = self.precision.get_integer_format()
# argument reduction
arg_reg_value = log(2)/2**index_size
inv_log2_value = round(1/arg_reg_value, self.precision.get_sollya_object(), sollya.RN)
inv_log2_cst = Constant(inv_log2_value, precision = self.precision, tag = "inv_log2")
# for r_hi to be accurate we ensure k * log2_hi_value_cst is exact
# by limiting the number of non-zero bits in log2_hi_value_cst
# cosh(x) ~ exp(abs(x))/2 for a big enough x
# cosh(x) > 2^1023 <=> exp(x) > 2^1024 <=> x > log(2^1024)
# k = inv_log2_value * x
# -1 for guard
max_k_approx = inv_log2_value * log(sollya.SollyaObject(2)**1024)
max_k_bitsize = int(ceil(log2(max_k_approx)))
Log.report(Log.Info, "max_k_bitsize: %d" % max_k_bitsize)
log2_hi_value_precision = self.precision.get_precision() - max_k_bitsize - 1
log2_hi_value = round(arg_reg_value, log2_hi_value_precision, sollya.RN)
log2_lo_value = round(arg_reg_value - log2_hi_value, self.precision.get_sollya_object(), sollya.RN)
log2_hi_value_cst = Constant(log2_hi_value, tag = "log2_hi_value", precision = self.precision)
log2_lo_value_cst = Constant(log2_lo_value, tag = "log2_lo_value", precision = self.precision)
k = Trunc(Multiplication(inv_log2_cst, vx), precision = self.precision)
k_log2 = Multiplication(k, log2_hi_value_cst, precision = self.precision, exact = True, tag = "k_log2", unbreakable = True)
r_hi = vx - k_log2
r_hi.set_attributes(tag = "r_hi", debug = debug_multi, unbreakable = True)
r_lo = -k * log2_lo_value_cst
# reduced argument
r = r_hi + r_lo
r.set_attributes(tag = "r", debug = debug_multi)
if is_gappa_installed():
r_eval_error = self.get_eval_error(r_hi, variable_copy_map =
{
vx: Variable("vx", interval = Interval(0, 715), precision = self.precision),
k: Variable("k", interval = Interval(0, 1024), precision = self.precision)
})
Log.report(Log.Verbose, "r_eval_error: ", r_eval_error)
approx_interval = Interval(-arg_reg_value, arg_reg_value)
error_goal_approx = 2**-(self.precision.get_precision())
poly_degree = sup(guessdegree(exp(sollya.x), approx_interval, error_goal_approx)) + 3
precision_list = [1] + [self.precision] * (poly_degree)
k_integer = Conversion(k, precision = int_precision, tag = "k_integer", debug = debug_multi)
k_hi = BitLogicRightShift(k_integer, Constant(index_size, precision=int_precision), tag = "k_int_hi", precision = int_precision, debug = debug_multi)
k_lo = Modulo(k_integer, 2**index_size, tag = "k_int_lo", precision = int_precision, debug = debug_multi)
pow_exp = ExponentInsertion(Conversion(k_hi, precision = int_precision), precision = self.precision, tag = "pow_exp", debug = debug_multi)
exp_table = ML_NewTable(dimensions = [2 * 2**index_size, 4], storage_precision = self.precision, tag = self.uniquify_name("exp2_table"))
for i in range(2 * 2**index_size):
input_value = i - 2**index_size if i >= 2**index_size else i
reduced_hi_prec = int(self.precision.get_mantissa_size() - 8)
# using SollyaObject wrapper to force evaluation by sollya
# with higher precision
exp_value = sollya.SollyaObject(2)**((input_value)* 2**-index_size)
mexp_value = sollya.SollyaObject(2)**((-input_value)* 2**-index_size)
pos_value_hi = round(exp_value, reduced_hi_prec, sollya.RN)
pos_value_lo = round(exp_value - pos_value_hi, self.precision.get_sollya_object(), sollya.RN)
neg_value_hi = round(mexp_value, reduced_hi_prec, sollya.RN)
neg_value_lo = round(mexp_value - neg_value_hi, self.precision.get_sollya_object(), sollya.RN)
exp_table[i][0] = neg_value_hi
exp_table[i][1] = neg_value_lo
exp_table[i][2] = pos_value_hi
exp_table[i][3] = pos_value_lo
# log2_value = log(2) / 2^index_size
# sinh(x) = 1/2 * (exp(x) - exp(-x))
# exp(x) = exp(x - k * log2_value + k * log2_value)
#
# r = x - k * log2_value
# exp(x) = exp(r) * 2 ^ (k / 2^index_size)
#
# k / 2^index_size = h + l * 2^-index_size, with k, h, l integers
# exp(x) = exp(r) * 2^h * 2^(l *2^-index_size)
#
# sinh(x) = exp(r) * 2^(h-1) * 2^(l *2^-index_size) - exp(-r) * 2^(-h-1) * 2^(-l *2^-index_size)
# S=2^(h-1), T = 2^(-h-1)
# exp(r) = 1 + poly_pos(r)
# exp(-r) = 1 + poly_neg(r)
# 2^(l / 2^index_size) = pos_value_hi + pos_value_lo
# 2^(-l / 2^index_size) = neg_value_hi + neg_value_lo
#
error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai)
poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(exp(sollya.x), poly_degree, precision_list, approx_interval, sollya.absolute, error_function = error_function)
Log.report(Log.Verbose, "poly_approx_error: {}, {}".format(poly_approx_error, float(log2(poly_approx_error))))
polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme
poly_pos = polynomial_scheme_builder(poly_object.sub_poly(start_index = 1), r, unified_precision = self.precision)
poly_pos.set_attributes(tag = "poly_pos", debug = debug_multi)
poly_neg = polynomial_scheme_builder(poly_object.sub_poly(start_index = 1), -r, unified_precision = self.precision)
poly_neg.set_attributes(tag = "poly_neg", debug = debug_multi)
table_index = Addition(k_lo, Constant(2**index_size, precision = int_precision), precision = int_precision, tag = "table_index", debug = debug_multi)
neg_value_load_hi = TableLoad(exp_table, table_index, 0, tag = "neg_value_load_hi", debug = debug_multi)
neg_value_load_lo = TableLoad(exp_table, table_index, 1, tag = "neg_value_load_lo", debug = debug_multi)
pos_value_load_hi = TableLoad(exp_table, table_index, 2, tag = "pos_value_load_hi", debug = debug_multi)
pos_value_load_lo = TableLoad(exp_table, table_index, 3, tag = "pos_value_load_lo", debug = debug_multi)
k_plus = Max(
Subtraction(k_hi, Constant(1, precision = int_precision), precision=int_precision, tag="k_plus", debug=debug_multi),
Constant(self.precision.get_emin_normal(), precision = int_precision))
k_neg = Max(
Subtraction(-k_hi, Constant(1, precision=int_precision), precision=int_precision, tag="k_neg", debug=debug_multi),
Constant(self.precision.get_emin_normal(), precision = int_precision))
# 2^(h-1)
pow_exp_pos = ExponentInsertion(k_plus, precision = self.precision, tag="pow_exp_pos", debug=debug_multi)
# 2^(-h-1)
pow_exp_neg = ExponentInsertion(k_neg, precision = self.precision, tag="pow_exp_neg", debug=debug_multi)
hi_terms = (pos_value_load_hi * pow_exp_pos - neg_value_load_hi * pow_exp_neg)
hi_terms.set_attributes(tag = "hi_terms", debug=debug_multi)
pos_exp = (pos_value_load_hi * poly_pos + (pos_value_load_lo + pos_value_load_lo * poly_pos)) * pow_exp_pos
pos_exp.set_attributes(tag = "pos_exp", debug = debug_multi)
neg_exp = (neg_value_load_hi * poly_neg + (neg_value_load_lo + neg_value_load_lo * poly_neg)) * pow_exp_neg
neg_exp.set_attributes(tag = "neg_exp", debug = debug_multi)
result = Addition(
Subtraction(
pos_exp,
neg_exp,
precision=self.precision,
),
hi_terms,
precision=self.precision,
tag="result",
debug=debug_multi
)
# ov_value
ov_value = round(asinh(self.precision.get_max_value()), self.precision.get_sollya_object(), sollya.RD)
ov_flag = Comparison(Abs(vx), Constant(ov_value, precision = self.precision), specifier = Comparison.Greater)
# main scheme
scheme = Statement(
Return(
Select(
ov_flag,
sign*FP_PlusInfty(self.precision),
sign*result
)))
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.get_input_precision().sollya_object
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# 2-limb approximation of log(2)
# hi part precision is reduced to provide exact operation
# when multiplied by an exponent value
log2_hi_value = round(log(2), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
int_precision = self.precision.get_integer_format()
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision = self.precision)
dummy_rcp_seed = ReciprocalSeed(dummy_var, precision = self.precision)
inv_approx_table = self.processor.get_recursive_implementation(dummy_rcp_seed, language = None, table_getter = lambda self: self.approx_table_map)
# table creation
table_index_size = inv_approx_table.index_size
log_table = ML_NewTable(dimensions = [2**table_index_size, 2], storage_precision = self.precision)
# storing accurate logarithm approximation of value returned
# by the fast reciprocal operation
for i in range(0, 2**table_index_size):
inv_value = inv_approx_table[i]
value_high = round(log(inv_value), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
neg_input = Comparison(vx, -1, likely=False, precision=ML_Bool, specifier=Comparison.Less, debug=debug_multi, tag="neg_input")
vx_nan_or_inf = Test(vx, specifier=Test.IsInfOrNaN, likely=False, precision=ML_Bool, debug=debug_multi, tag="nan_or_inf")
vx_snan = Test(vx, specifier=Test.IsSignalingNaN, likely=False, debug=debug_multi, tag="snan")
vx_inf = Test(vx, specifier=Test.IsInfty, likely=False, debug=debug_multi, tag="inf")
vx_subnormal = Test(vx, specifier=Test.IsSubnormal, likely=False, debug=debug_multi, tag="vx_subnormal")
# for x = m.2^e, such that e >= 0
#
# log(1+x) = log(1 + m.2^e)
# = log(2^e . 2^-e + m.2^e)
# = log(2^e . (2^-e + m))
# = log(2^e) + log(2^-e + m)
# = e . log(2) + log (2^-e + m)
#
# t = (2^-e + m)
# t = m_t . 2^e_t
# r ~ 1 / m_t => r.m_t ~ 1 ~ 0
#
# t' = t . 2^-e_t
# = 2^-e-e_t + m . 2^-e_t
#
# if e >= 0, then 2^-e <= 1, then 1 <= m + 2^-e <= 3
# r = m_r . 2^e_r
#
# log(1+x) = e.log(2) + log(r . 2^e_t . 2^-e_t . (2^-e + m) / r)
# = e.log(2) + log(r . 2^(-e-e_t) + r.m.2^-e_t) + e_t . log(2)- log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + P_log1p(r . t' - 1) - log(r)
#
#
# argument reduction
m = MantissaExtraction(vx, tag="vx", precision=self.precision, debug=debug_multi)
e = ExponentExtraction(vx, tag="e", precision=int_precision, debug=debug_multi)
# 2^-e
TwoMinusE = ExponentInsertion(-e, tag="Two_minus_e", precision=self.precision, debug=debug_multi)
t = Addition(TwoMinusE, m, precision=self.precision, tag="t", debug=debug_multi)
m_t = MantissaExtraction(t, tag="m_t", precision=self.precision, debug=debug_multi)
e_t = ExponentExtraction(t, tag="e_t", precision=int_precision, debug=debug_multi)
# 2^(-e-e_t)
TwoMinusEEt = ExponentInsertion(-e-e_t, tag="Two_minus_e_et", precision=self.precision)
TwoMinusEt = ExponentInsertion(-e_t, tag="Two_minus_et", precision=self.precision, debug=debug_multi)
rcp_mt = ReciprocalSeed(m_t, tag="rcp_mt", precision=self.precision, debug=debug_multi)
INDEX_SIZE = table_index_size
table_index = generic_mantissa_msb_index_fct(INDEX_SIZE, m_t)
table_index.set_attributes(tag="table_index", debug=debug_multi)
log_inv_lo = TableLoad(log_table, table_index, 1, tag="log_inv_lo", debug=debug_multi)
log_inv_hi = TableLoad(log_table, table_index, 0, tag="log_inv_hi", debug=debug_multi)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
approx_fct = sollya.log1p(sollya.x) / (sollya.x)
poly_degree = sup(guessdegree(approx_fct, approx_interval, S2**-(self.precision.get_field_size()+1))) + 1
Log.report(Log.Debug, "poly_degree is {}", poly_degree)
global_poly_object = Polynomial.build_from_approximation(approx_fct, poly_degree, [self.precision]*(poly_degree+1), approx_interval, sollya.absolute)
poly_object = global_poly_object # .sub_poly(start_index=1)
EXT_PRECISION_MAP = {
ML_Binary32: ML_SingleSingle,
ML_Binary64: ML_DoubleDouble,
ML_SingleSingle: ML_TripleSingle,
ML_DoubleDouble: ML_TripleDouble
}
if not self.precision in EXT_PRECISION_MAP:
Log.report(Log.Error, "no extended precision available for {}", self.precision)
ext_precision = EXT_PRECISION_MAP[self.precision]
# pre_rtp = r . 2^(-e-e_t) + m .2^-e_t
pre_rtp = Addition(
rcp_mt * TwoMinusEEt,
Multiplication(
rcp_mt,
Multiplication(
m,
TwoMinusEt,
precision=self.precision,
tag="pre_mult",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_mult2",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_rtp",
debug=debug_multi
)
pre_red_vx = Addition(
pre_rtp,
-1,
precision=ext_precision,
)
red_vx = Conversion(pre_red_vx, precision=self.precision, tag="red_vx", debug=debug_multi)
Log.report(Log.Info, "generating polynomial evaluation scheme")
poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vx, unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Debug, "{}", global_poly_object.get_sollya_object())
fp_e = Conversion(e + e_t, precision=self.precision, tag="fp_e", debug=debug_multi)
ext_poly = Multiplication(red_vx, poly, precision=ext_precision)
pre_result = Addition(
Addition(
fp_e * log2_hi,
fp_e * log2_lo,
precision=ext_precision
),
Addition(
Addition(
-log_inv_hi,
-log_inv_lo,
precision=ext_precision
),
ext_poly,
precision=ext_precision
),
precision=ext_precision
)
result = Conversion(pre_result, precision=self.precision, tag="result", debug=debug_multi)
# main scheme
Log.report(Log.Info, "MDL scheme")
pre_scheme = ConditionBlock(neg_input,
Statement(
ClearException(),
Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))
),
ConditionBlock(vx_nan_or_inf,
ConditionBlock(vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(
ClearException(),
ConditionBlock(vx_snan,
Raise(ML_FPE_Invalid)
),
Return(FP_QNaN(self.precision))
)
),
Return(result)
)
)
scheme = pre_scheme
return scheme
def generic_poly_split(offset_fct, indexing, target_eps, coeff_precision, vx):
""" generate the meta approximation for @p offset_fct over several
intervals defined by @p indexing object
For each sub-interval, a polynomial approximation with
maximal_error @p target_eps is tabulated, and evaluated using format
@p coeff_precision.
The input variable is @p vx """
# computing degree for a different polynomial approximation on each
# sub-interval
poly_degree_list = [
int(sup(guessdegree(offset_fct(offset), sub_interval, target_eps)))
for offset, sub_interval in indexing.get_offseted_sub_list()
]
poly_max_degree = max(poly_degree_list)
# tabulating polynomial coefficients on split_num sub-interval of interval
poly_table = ML_NewTable(
dimensions=[indexing.split_num, poly_max_degree + 1],
storage_precision=coeff_precision,
const=True)
offset_table = ML_NewTable(dimensions=[indexing.split_num],
storage_precision=coeff_precision,
const=True)
max_error = 0.0
for sub_index in range(indexing.split_num):
poly_degree = poly_degree_list[sub_index]
offset, approx_interval = indexing.get_offseted_sub_interval(sub_index)
offset_table[sub_index] = offset
if poly_degree == 0:
# managing constant approximation separately since it seems
# to break sollya
local_approx = coeff_precision.round_sollya_object(
offset_fct(offset)(inf(approx_interval)))
poly_table[sub_index][0] = local_approx
for monomial_index in range(1, poly_max_degree + 1):
poly_table[sub_index][monomial_index] = 0
approx_error = sollya.infnorm(
offset_fct(offset) - local_approx, approx_interval)
else:
poly_object, approx_error = Polynomial.build_from_approximation_with_error(
offset_fct(offset), poly_degree,
[coeff_precision] * (poly_degree + 1), approx_interval,
sollya.relative)
for monomial_index in range(poly_max_degree + 1):
if monomial_index <= poly_degree:
poly_table[sub_index][
monomial_index] = poly_object.coeff_map[monomial_index]
else:
poly_table[sub_index][monomial_index] = 0
max_error = max(approx_error, max_error)
Log.report(Log.Debug, "max approx error is {}", max_error)
# indexing function: derive index from input @p vx value
poly_index = indexing.get_index_node(vx)
poly_index.set_attributes(tag="poly_index", debug=debug_multi)
ext_precision = get_extended_fp_precision(coeff_precision)
# building polynomial evaluation scheme
offset = TableLoad(offset_table,
poly_index,
precision=coeff_precision,
tag="offset",
debug=debug_multi)
poly = TableLoad(poly_table,
poly_index,
poly_max_degree,
precision=coeff_precision,
tag="poly_init",
debug=debug_multi)
red_vx = Subtraction(vx,
offset,
precision=vx.precision,
tag="red_vx",
debug=debug_multi)
for monomial_index in range(poly_max_degree, -1, -1):
coeff = TableLoad(poly_table,
poly_index,
monomial_index,
precision=coeff_precision,
tag="poly_%d" % monomial_index,
debug=debug_multi)
#fma_precision = coeff_precision if monomial_index > 1 else ext_precision
fma_precision = coeff_precision
poly = FMA(red_vx, poly, coeff, precision=fma_precision)
#return Conversion(poly, precision=coeff_precision)
#return poly.hi
return poly
def generate_scheme(self):
# declaring main input variable
vx = self.implementation.add_input_variable("x", self.precision)
# declaring approximation parameters
index_size = 6
num_iteration = 8
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)
def cbrt_newton_iteration(current_approx, input_value, input_inverse):
# Cubic root of A is approximated by a Newton-Raphson iteration
# on f(x) = 1 - A / x^3
# x_n+1 = 4/3 * x_n - x_n^4 / (3 * A)
# x_n+1 = 1/3 * (x_n * (1 - x_n^3/A) + x_n)
approx_triple = Multiplication(
current_approx, Multiplication(current_approx, current_approx))
diff = FMSN(approx_triple, input_inverse,
Constant(1, precision=self.precision))
injection = FMA(
Multiplication(
current_approx,
Constant(1 / 3.0, precision=self.precision),
), diff, current_approx)
new_approx = injection
return new_approx
reduced_vx = MantissaExtraction(vx, precision=self.precision)
int_precision = self.precision.get_integer_format()
cbrt_approx_table = ML_NewTable(
dimensions=[2**index_size, 1],
storage_precision=self.precision,
tag=self.uniquify_name("cbrt_approx_table"))
for i in range(2**index_size):
input_value = 1 + i / SollyaObject(2**index_size)
cbrt_approx = cbrt(input_value)
cbrt_approx_table[i][0] = round(cbrt_approx,
self.precision.get_sollya_object(),
RN)
# Modulo operations will returns a reduced exponent within [-3, 2]
# so we approximate cbrt on this interval (with index offset by -3)
cbrt_mod_table = ML_NewTable(dimensions=[6, 1],
storage_precision=self.precision,
tag=self.uniquify_name("cbrt_mod_table"))
for i in range(6):
input_value = SollyaObject(2)**(i - 3)
cbrt_mod_table[i][0] = round(cbrt(input_value),
self.precision.get_sollya_object(),
RN)
vx_int = TypeCast(reduced_vx, precision=int_precision)
mask = BitLogicRightShift(vx_int,
self.precision.get_precision() - index_size,
precision=int_precision)
mask = BitLogicAnd(mask,
Constant(2**index_size - 1,
precision=int_precision),
precision=int_precision,
tag="table_index",
debug=debug_multi)
table_index = mask
int_precision = self.precision.get_integer_format()
exp_vx = ExponentExtraction(vx, precision=int_precision, tag="exp_vx")
exp_vx_third = Division(exp_vx,
Constant(3, precision=int_precision),
precision=int_precision,
tag="exp_vx_third")
exp_vx_mod = Modulo(exp_vx,
Constant(3, precision=int_precision),
precision=int_precision,
tag="exp_vx_mod",
debug=debug_multi)
# offset on modulo to make sure table index is positive
exp_vx_mod = exp_vx_mod + 3
cbrt_mod = TableLoad(cbrt_mod_table,
exp_vx_mod,
Constant(0),
tag="cbrt_mod")
init_approx = Multiplication(
Multiplication(
# approx cbrt(mantissa)
TableLoad(cbrt_approx_table,
table_index,
Constant(0, precision=ML_Int32),
tag="seed",
debug=debug_multi),
# approx cbrt(2^(e%3))
cbrt_mod,
tag="init_mult",
debug=debug_multi,
precision=self.precision),
# 2^(e/3)
ExponentInsertion(exp_vx_third,
precision=self.precision,
tag="exp_vx_third",
debug=debug_multi),
tag="init_approx",
debug=debug_multi,
precision=self.precision)
inverse_red_vx = Division(Constant(1, precision=self.precision),
reduced_vx)
inverse_vx = Division(Constant(1, precision=self.precision), vx)
current_approx = init_approx
for i in range(num_iteration):
#current_approx = cbrt_newton_iteration(current_approx, reduced_vx, inverse_red_vx)
current_approx = cbrt_newton_iteration(current_approx, vx,
inverse_vx)
current_approx.set_attributes(tag="approx_%d" % i,
debug=debug_multi)
result = current_approx
result.set_attributes(tag="result", debug=debug_multi)
# last iteration
ext_precision = ML_DoubleDouble
xn_2 = Multiplication(current_approx,
current_approx,
precision=ext_precision)
xn_3 = Multiplication(current_approx, xn_2, precision=ext_precision)
FourThird = Constant(4 / SollyaObject(3), precision=ext_precision)
# main scheme
Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m")
scheme = Statement(Return(result))
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x",
self.get_input_precision())
sollya_precision = self.get_input_precision().get_sollya_object()
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# testing special value inputs
test_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=True,
tag="nan_or_inf")
test_nan = Test(vx,
specifier=Test.IsNaN,
debug=True,
tag="is_nan_test")
test_positive = Comparison(vx,
0,
specifier=Comparison.GreaterOrEqual,
debug=True,
tag="inf_sign")
test_signaling_nan = Test(vx,
specifier=Test.IsSignalingNaN,
debug=True,
tag="is_signaling_nan")
# if input is a signaling NaN, raise an invalid exception and returns
# a quiet NaN
return_snan = Statement(
ExpRaiseReturn(ML_FPE_Invalid,
return_value=FP_QNaN(self.precision)))
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = self.precision.get_integer_format()
# log2(vx)
# r = vx_mant
# e = vx_exp
# vx reduced to r in [1, 2[
# log2(vx) = log2(r * 2^e)
# = log2(r) + e
#
## log2(r) is approximated by
# log2(r) = log2(inv_seed(r) * r / inv_seed(r)
# = log2(inv_seed(r) * r) - log2(inv_seed(r))
# inv_seed(r) in ]1/2, 1] => log2(inv_seed(r)) in ]-1, 0]
#
# inv_seed(r) * r ~ 1
# we can easily tabulate -log2(inv_seed(r))
#
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
log_table = ML_NewTable(dimensions=[2**table_index_size, 2],
storage_precision=self.precision,
tag=self.uniquify_name("inv_table"))
# value for index 0 is set to 0.0
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in range(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
#inv_value = (1.0 + (inv_approx_table[i][0] / S2**9) ) * S2**-1
#print inv_approx_table[i][0], inv_value
inv_value = inv_approx_table[i][0]
value_high_bitsize = self.precision.get_field_size() - (
self.precision.get_exponent_size() + 1)
value_high = round(log2(inv_value), value_high_bitsize, sollya.RN)
value_low = round(
log2(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
def compute_log(_vx, exp_corr_factor=None):
_vx_mant = MantissaExtraction(_vx,
tag="_vx_mant",
precision=self.precision,
debug=debug_lftolx)
_vx_exp = ExponentExtraction(_vx, tag="_vx_exp", debug=debugd)
# The main table is indexed by the 7 most significant bits
# of the mantissa
table_index = inv_approx_table.index_function(_vx_mant)
table_index.set_attributes(tag="table_index", debug=debuglld)
# argument reduction
# Using AND -2 to exclude LSB set to 1 for Newton-Raphson convergence
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(
TypeCast(DivisionSeed(_vx_mant,
precision=self.precision,
tag="seed",
debug=debug_lftolx,
silent=True),
precision=ML_UInt64),
Constant(-2, precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
1.0,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
_red_vx = FMA(arg_red_index, _vx_mant, -1.0)
_red_vx.set_attributes(tag="_red_vx", debug=debug_lftolx)
inv_err = S2**-inv_approx_table.index_size
red_interval = Interval(1 - inv_err, 1 + inv_err)
# return in case of standard (non-special) input
_log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
_log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
Log.report(Log.Verbose, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log2(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() * 1.1))) + 1
sollya.settings.display = sollya.hexadecimal
global_poly_object, approx_error = Polynomial.build_from_approximation_with_error(
log2(1 + sollya.x) / sollya.x,
poly_degree, [self.precision] * (poly_degree + 1),
approx_interval,
sollya.absolute,
error_function=lambda p, f, ai, mod, t: sollya.dirtyinfnorm(
p - f, ai))
Log.report(
Log.Info, "poly_degree={}, approx_error={}".format(
poly_degree, approx_error))
poly_object = global_poly_object.sub_poly(start_index=1, offset=1)
#poly_object = global_poly_object.sub_poly(start_index=0,offset=0)
Attributes.set_default_silent(True)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Log.report(Log.Verbose, "generating polynomial evaluation scheme")
pre_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, _red_vx, unified_precision=self.precision)
_poly = FMA(pre_poly, _red_vx,
global_poly_object.get_cst_coeff(0, self.precision))
_poly.set_attributes(tag="poly", debug=debug_lftolx)
Log.report(
Log.Verbose, "sollya global_poly_object: {}".format(
global_poly_object.get_sollya_object()))
Log.report(
Log.Verbose, "sollya poly_object: {}".format(
poly_object.get_sollya_object()))
corr_exp = _vx_exp if exp_corr_factor == None else _vx_exp + exp_corr_factor
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
pre_result = -_log_inv_hi + (_red_vx * _poly + (-_log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = Conversion(corr_exp, precision=self.precision)
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_hex",
debug=debug_lftolx)
_result = exact_log2_hi_exp + pre_result
return _result, _poly, _log_inv_lo, _log_inv_hi, _red_vx
result, poly, log_inv_lo, log_inv_hi, red_vx = compute_log(vx)
result.set_attributes(tag="result", debug=debug_lftolx)
# specific input value predicate
neg_input = Comparison(vx,
0,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="vx_snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="vx_inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
vx_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
debug=debugd,
tag="vx_zero")
exp_mone = Equal(vx_exp,
-1,
tag="exp_minus_one",
debug=debugd,
likely=False)
vx_one = Equal(vx, 1.0, tag="vx_one", likely=False, debug=debugd)
# Specific specific for the case exp == -1
# log2(x) = log2(m) - 1
#
# as m in [1, 2[, log2(m) in [0, 1[
# if r is close to 2, a catastrophic cancellation can occur
#
# r = seed(m)
# log2(x) = log2(seed(m) * m / seed(m)) - 1
# = log2(seed(m) * m) - log2(seed(m)) - 1
#
# for m really close to 2 => seed(m) = 0.5
# => log2(x) = log2(0.5 * m)
# =
result_exp_m1 = (-log_inv_hi - 1.0) + FMA(poly, red_vx, -log_inv_lo)
result_exp_m1.set_attributes(tag="result_exp_m1", debug=debug_lftolx)
m100 = -100
S2100 = Constant(S2**100, precision=self.precision)
result_subnormal, _, _, _, _ = compute_log(vx * S2100,
exp_corr_factor=m100)
result_subnormal.set_attributes(tag="result_subnormal",
debug=debug_lftolx)
one_err = S2**-7
approx_interval_one = Interval(-one_err, one_err)
red_vx_one = vx - 1.0
poly_degree_one = sup(
guessdegree(
log(1 + x) / x, approx_interval_one, S2**
-(self.precision.get_field_size() + 1))) + 1
poly_object_one = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree_one,
[self.precision] * (poly_degree_one + 1), approx_interval_one,
absolute).sub_poly(start_index=1)
poly_one = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object_one, red_vx_one, unified_precision=self.precision)
poly_one.set_attributes(tag="poly_one", debug=debug_lftolx)
result_one = red_vx_one + red_vx_one * poly_one
cond_one = (vx < (1 + one_err)) & (vx > (1 - one_err))
cond_one.set_attributes(tag="cond_one", debug=debugd, likely=False)
# main scheme
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal,
ConditionBlock(
vx_zero,
Statement(
ClearException(),
Raise(ML_FPE_DivideByZero),
Return(FP_MinusInfty(self.precision)),
),
Statement(ClearException(), result_subnormal,
Return(result_subnormal))),
ConditionBlock(
vx_one,
Statement(
ClearException(),
Return(FP_PlusZero(self.precision)),
),
ConditionBlock(exp_mone, Return(result_exp_m1),
Return(result))))))
scheme = Statement(result, pre_scheme)
return scheme
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.get_input_precision().sollya_object
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
log2_hi_value = round(log(2), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision = self.precision)
log2_lo = Constant(log2_lo_value, precision = self.precision)
vx_exp = ExponentExtraction(vx, tag = "vx_exp", debug = debugd)
int_precision = self.precision.get_integer_format()
# 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 = 7
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 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] / S2**9) ) * S2**-1
value_high = round(log(inv_value), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
vx_exp = ExponentExtraction(vx, tag = "vx_exp", debug = debugd)
# case close to 0: ctz
ctz_exp_limit = -7
ctz_cond = vx_exp < ctz_exp_limit
ctz_interval = Interval(-S2**ctz_exp_limit, S2**ctz_exp_limit)
ctz_poly_degree = sup(guessdegree(log1p(sollya.x)/sollya.x, ctz_interval, S2**-(self.precision.get_field_size()+1))) + 1
ctz_poly_object = Polynomial.build_from_approximation(log1p(sollya.x)/sollya.x, ctz_poly_degree, [self.precision]*(ctz_poly_degree+1), ctz_interval, sollya.absolute)
Log.report(Log.Info, "generating polynomial evaluation scheme")
ctz_poly = PolynomialSchemeEvaluator.generate_horner_scheme(ctz_poly_object, vx, unified_precision = self.precision)
ctz_poly.set_attributes(tag = "ctz_poly", debug = debug_lftolx)
ctz_result = vx * ctz_poly
neg_input = Comparison(vx, -1, likely = False, specifier = Comparison.Less, debug = debugd, tag = "neg_input")
vx_nan_or_inf = Test(vx, specifier = Test.IsInfOrNaN, likely = False, debug = debugd, tag = "nan_or_inf")
vx_snan = Test(vx, specifier = Test.IsSignalingNaN, likely = False, debug = debugd, tag = "snan")
vx_inf = Test(vx, specifier = Test.IsInfty, likely = False, debug = debugd, tag = "inf")
vx_subnormal = Test(vx, specifier = Test.IsSubnormal, likely = False, debug = debugd, tag = "vx_subnormal")
log_function_code = CodeFunction("new_log", [Variable("x", precision = ML_Binary64)], output_format = ML_Binary64)
log_call_generator = FunctionOperator(log_function_code.get_name(), arity = 1, output_precision = ML_Binary64, declare_prototype = log_function_code)
newlog_function = FunctionObject(log_function_code.get_name(), (ML_Binary64,), ML_Binary64, log_call_generator)
# case away from 0.0
pre_vxp1 = vx + 1.0
pre_vxp1.set_attributes(tag = "pre_vxp1", debug = debug_lftolx)
pre_vxp1_exp = ExponentExtraction(pre_vxp1, tag = "pre_vxp1_exp", debug = debugd)
cm500 = Constant(-500, precision = ML_Int32)
c0 = Constant(0, precision = ML_Int32)
cond_scaling = pre_vxp1_exp > 2**(self.precision.get_exponent_size()-2)
scaling_factor_exp = Select(cond_scaling, cm500, c0)
scaling_factor = ExponentInsertion(scaling_factor_exp, precision = self.precision, tag = "scaling_factor")
vxp1 = pre_vxp1 * scaling_factor
vxp1.set_attributes(tag = "vxp1", debug = debug_lftolx)
vxp1_exp = ExponentExtraction(vxp1, tag = "vxp1_exp", debug = debugd)
vxp1_inv = ReciprocalSeed(vxp1, precision = self.precision, tag = "vxp1_inv", debug = debug_lftolx, silent = True)
vxp1_dirty_inv = ExponentInsertion(-vxp1_exp, precision = self.precision, tag = "vxp1_dirty_inv", debug = debug_lftolx)
table_index = BitLogicAnd(BitLogicRightShift(TypeCast(vxp1, precision = int_precision, debug = debuglx), self.precision.get_field_size() - 7, debug = debuglx), 0x7f, tag = "table_index", debug = debuglx)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(TypeCast(vxp1_inv, precision = ML_UInt64), Constant(-2, precision = ML_UInt64), precision = ML_UInt64), precision = self.precision, tag = "pre_arg_red_index", debug = debug_lftolx)
arg_red_index = Select(Equal(table_index, 0), vxp1_dirty_inv, pre_arg_red_index, tag = "arg_red_index", debug = debug_lftolx)
red_vxp1 = Select(cond_scaling, arg_red_index * vxp1 - 1.0, (arg_red_index * vx - 1.0) + arg_red_index)
#red_vxp1 = arg_red_index * vxp1 - 1.0
red_vxp1.set_attributes(tag = "red_vxp1", debug = debug_lftolx)
log_inv_lo = TableLoad(log_table, table_index, 1, tag = "log_inv_lo", debug = debug_lftolx)
log_inv_hi = TableLoad(log_table, table_index, 0, tag = "log_inv_hi", debug = debug_lftolx)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
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+sollya.x)/sollya.x, poly_degree, [self.precision]*(poly_degree+1), approx_interval, sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index = 1)
Log.report(Log.Info, "generating polynomial evaluation scheme")
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(poly_object, red_vxp1, unified_precision = self.precision)
_poly.set_attributes(tag = "poly", debug = debug_lftolx)
Log.report(Log.Info, global_poly_object.get_sollya_object())
vxp1_inv_exp = ExponentExtraction(vxp1_inv, tag = "vxp1_inv_exp", debug = debugd)
corr_exp = Conversion(-vxp1_exp + scaling_factor_exp, precision = self.precision)# vxp1_inv_exp
#poly = (red_vxp1) * (1 + _poly)
#poly.set_attributes(tag = "poly", debug = debug_lftolx, prevent_optimization = True)
pre_result = -log_inv_hi + (red_vxp1 + red_vxp1 * _poly + (-corr_exp * log2_lo - log_inv_lo))
pre_result.set_attributes(tag = "pre_result", debug = debug_lftolx)
exact_log2_hi_exp = - corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag = "exact_log2_hi_exp", debug = debug_lftolx, prevent_optimization = True)
#std_result = exact_log2_hi_exp + pre_result
exact_log2_lo_exp = - corr_exp * log2_lo
exact_log2_lo_exp.set_attributes(tag = "exact_log2_lo_exp", debug = debug_lftolx)#, prevent_optimization = True)
init = exact_log2_lo_exp - log_inv_lo
init.set_attributes(tag = "init", debug = debug_lftolx, prevent_optimization = True)
fma0 = (red_vxp1 * _poly + init) # - log_inv_lo)
fma0.set_attributes(tag = "fma0", debug = debug_lftolx)
step0 = fma0
step0.set_attributes(tag = "step0", debug = debug_lftolx) #, prevent_optimization = True)
step1 = step0 + red_vxp1
step1.set_attributes(tag = "step1", debug = debug_lftolx, prevent_optimization = True)
step2 = -log_inv_hi + step1
step2.set_attributes(tag = "step2", debug = debug_lftolx, prevent_optimization = True)
std_result = exact_log2_hi_exp + step2
std_result.set_attributes(tag = "std_result", debug = debug_lftolx, prevent_optimization = True)
# main scheme
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,
Return(vx),
ConditionBlock(ctz_cond,
Statement(
Return(ctz_result),
),
Statement(
Return(std_result)
)
)
)
)
)
scheme = pre_scheme
return scheme