def __init__(self,
precision=ML_Binary32,
abs_accuracy=S2**-24,
libm_compliant=True,
debug_flag=False,
fuse_fma=True,
fast_path_extract=True,
target=GenericProcessor(),
output_file="log1pf.c",
function_name="log1pf"):
# declaring CodeFunction and retrieving input variable
self.function_name = function_name
self.precision = precision
self.processor = target
func_implementation = CodeFunction(self.function_name,
output_format=self.precision)
vx = func_implementation.add_input_variable("x", self.precision)
sollya_precision = self.precision.sollya_object
# debug utilities
debugf = ML_Debug(display_format="%f")
debuglf = ML_Debug(display_format="%lf")
debugx = ML_Debug(display_format="%x")
debuglx = ML_Debug(display_format="%\"PRIx64\"", )
debugd = ML_Debug(display_format="%d",
pre_process=lambda v: "(int) %s" % v)
debugld = ML_Debug(display_format="%ld")
#debug_lftolx = ML_Debug(display_format = "%\"PRIx64\"", pre_process = lambda v: "double_to_64b_encoding(%s)" % v)
debug_lftolx = ML_Debug(
display_format="%\"PRIx64\" ev=%x",
pre_process=lambda v:
"double_to_64b_encoding(%s), __k1_fpu_get_exceptions()" % v)
debug_ddtolx = ML_Debug(
display_format="%\"PRIx64\" %\"PRIx64\"",
pre_process=lambda v:
"double_to_64b_encoding(%s.hi), double_to_64b_encoding(%s.lo)" %
(v, v))
debug_dd = ML_Debug(display_format="{.hi=%lf, .lo=%lf}",
pre_process=lambda v: "%s.hi, %s.lo" % (v, v))
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
log2_hi_value = round(
log(2),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(
log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
int_precision = ML_Int64 if self.precision is ML_Binary64 else ML_Int32
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = DivisionSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
# table creation
table_index_size = 7
log_table = ML_Table(dimensions=[2**table_index_size, 2],
storage_precision=self.precision)
log_table[0][0] = 0.0
log_table[0][1] = 0.0
for i in xrange(1, 2**table_index_size):
#inv_value = (1.0 + (self.processor.inv_approx_table[i] / S2**9) + S2**-52) * S2**-1
inv_value = (1.0 + (inv_approx_table[i][0] / S2**9)) * S2**-1
value_high = round(
log(inv_value),
self.precision.get_field_size() -
(self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(
log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
vx_exp = ExponentExtraction(vx, tag="vx_exp", debug=debugd)
# case close to 0: ctz
ctz_exp_limit = -7
ctz_cond = vx_exp < ctz_exp_limit
ctz_interval = Interval(-S2**ctz_exp_limit, S2**ctz_exp_limit)
ctz_poly_degree = sup(
guessdegree(
log1p(sollya.x) / sollya.x, ctz_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
ctz_poly_object = Polynomial.build_from_approximation(
log1p(sollya.x) / sollya.x, ctz_poly_degree,
[self.precision] * (ctz_poly_degree + 1), ctz_interval,
sollya.absolute)
print "generating polynomial evaluation scheme"
ctz_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
ctz_poly_object, vx, unified_precision=self.precision)
ctz_poly.set_attributes(tag="ctz_poly", debug=debug_lftolx)
ctz_result = vx * ctz_poly
neg_input = Comparison(vx,
-1,
likely=False,
specifier=Comparison.Less,
debug=debugd,
tag="neg_input")
vx_nan_or_inf = Test(vx,
specifier=Test.IsInfOrNaN,
likely=False,
debug=debugd,
tag="nan_or_inf")
vx_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
debug=debugd,
tag="snan")
vx_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
debug=debugd,
tag="inf")
vx_subnormal = Test(vx,
specifier=Test.IsSubnormal,
likely=False,
debug=debugd,
tag="vx_subnormal")
log_function_code = CodeFunction(
"new_log", [Variable("x", precision=ML_Binary64)],
output_format=ML_Binary64)
log_call_generator = FunctionOperator(
log_function_code.get_name(),
arity=1,
output_precision=ML_Binary64,
declare_prototype=log_function_code)
newlog_function = FunctionObject(log_function_code.get_name(),
(ML_Binary64, ), ML_Binary64,
log_call_generator)
# case away from 0.0
pre_vxp1 = vx + 1.0
pre_vxp1.set_attributes(tag="pre_vxp1", debug=debug_lftolx)
pre_vxp1_exp = ExponentExtraction(pre_vxp1,
tag="pre_vxp1_exp",
debug=debugd)
cm500 = Constant(-500, precision=ML_Int32)
c0 = Constant(0, precision=ML_Int32)
cond_scaling = pre_vxp1_exp > 2**(self.precision.get_exponent_size() -
2)
scaling_factor_exp = Select(cond_scaling, cm500, c0)
scaling_factor = ExponentInsertion(scaling_factor_exp,
precision=self.precision,
tag="scaling_factor")
vxp1 = pre_vxp1 * scaling_factor
vxp1.set_attributes(tag="vxp1", debug=debug_lftolx)
vxp1_exp = ExponentExtraction(vxp1, tag="vxp1_exp", debug=debugd)
vxp1_inv = DivisionSeed(vxp1,
precision=self.precision,
tag="vxp1_inv",
debug=debug_lftolx,
silent=True)
vxp1_dirty_inv = ExponentInsertion(-vxp1_exp,
precision=self.precision,
tag="vxp1_dirty_inv",
debug=debug_lftolx)
table_index = BitLogicAnd(BitLogicRightShift(
TypeCast(vxp1, precision=int_precision, debug=debuglx),
self.precision.get_field_size() - 7,
debug=debuglx),
0x7f,
tag="table_index",
debug=debuglx)
# argument reduction
# TODO: detect if single operand inverse seed is supported by the targeted architecture
pre_arg_red_index = TypeCast(BitLogicAnd(TypeCast(vxp1_inv,
precision=ML_UInt64),
Constant(-2,
precision=ML_UInt64),
precision=ML_UInt64),
precision=self.precision,
tag="pre_arg_red_index",
debug=debug_lftolx)
arg_red_index = Select(Equal(table_index, 0),
vxp1_dirty_inv,
pre_arg_red_index,
tag="arg_red_index",
debug=debug_lftolx)
red_vxp1 = Select(cond_scaling, arg_red_index * vxp1 - 1.0,
(arg_red_index * vx - 1.0) + arg_red_index)
#red_vxp1 = arg_red_index * vxp1 - 1.0
red_vxp1.set_attributes(tag="red_vxp1", debug=debug_lftolx)
log_inv_lo = TableLoad(log_table,
table_index,
1,
tag="log_inv_lo",
debug=debug_lftolx)
log_inv_hi = TableLoad(log_table,
table_index,
0,
tag="log_inv_hi",
debug=debug_lftolx)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
print "building mathematical polynomial"
approx_interval = Interval(-inv_err, inv_err)
poly_degree = sup(
guessdegree(
log(1 + sollya.x) / sollya.x, approx_interval, S2**
-(self.precision.get_field_size() + 1))) + 1
global_poly_object = Polynomial.build_from_approximation(
log(1 + sollya.x) / sollya.x, poly_degree,
[self.precision] * (poly_degree + 1), approx_interval,
sollya.absolute)
poly_object = global_poly_object.sub_poly(start_index=1)
print "generating polynomial evaluation scheme"
_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vxp1, unified_precision=self.precision)
_poly.set_attributes(tag="poly", debug=debug_lftolx)
print global_poly_object.get_sollya_object()
vxp1_inv_exp = ExponentExtraction(vxp1_inv,
tag="vxp1_inv_exp",
debug=debugd)
corr_exp = -vxp1_exp + scaling_factor_exp # vxp1_inv_exp
#poly = (red_vxp1) * (1 + _poly)
#poly.set_attributes(tag = "poly", debug = debug_lftolx, prevent_optimization = True)
pre_result = -log_inv_hi + (red_vxp1 + red_vxp1 * _poly +
(-corr_exp * log2_lo - log_inv_lo))
pre_result.set_attributes(tag="pre_result", debug=debug_lftolx)
exact_log2_hi_exp = -corr_exp * log2_hi
exact_log2_hi_exp.set_attributes(tag="exact_log2_hi_exp",
debug=debug_lftolx,
prevent_optimization=True)
#std_result = exact_log2_hi_exp + pre_result
exact_log2_lo_exp = -corr_exp * log2_lo
exact_log2_lo_exp.set_attributes(
tag="exact_log2_lo_exp",
debug=debug_lftolx) #, prevent_optimization = True)
init = exact_log2_lo_exp - log_inv_lo
init.set_attributes(tag="init",
debug=debug_lftolx,
prevent_optimization=True)
fma0 = (red_vxp1 * _poly + init) # - log_inv_lo)
fma0.set_attributes(tag="fma0", debug=debug_lftolx)
step0 = fma0
step0.set_attributes(
tag="step0", debug=debug_lftolx) #, prevent_optimization = True)
step1 = step0 + red_vxp1
step1.set_attributes(tag="step1",
debug=debug_lftolx,
prevent_optimization=True)
step2 = -log_inv_hi + step1
step2.set_attributes(tag="step2",
debug=debug_lftolx,
prevent_optimization=True)
std_result = exact_log2_hi_exp + step2
std_result.set_attributes(tag="std_result",
debug=debug_lftolx,
prevent_optimization=True)
# main scheme
print "MDL scheme"
pre_scheme = ConditionBlock(
neg_input,
Statement(ClearException(), Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))),
ConditionBlock(
vx_nan_or_inf,
ConditionBlock(
vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(ClearException(),
ConditionBlock(vx_snan, Raise(ML_FPE_Invalid)),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
vx_subnormal, Return(vx),
ConditionBlock(ctz_cond, Statement(Return(ctz_result), ),
Statement(Return(std_result))))))
scheme = pre_scheme
#print scheme.get_str(depth = None, display_precision = True)
opt_eng = OptimizationEngine(self.processor)
# fusing FMA
print "MDL fusing FMA"
scheme = opt_eng.fuse_multiply_add(scheme, silence=True)
print "MDL abstract scheme"
opt_eng.instantiate_abstract_precision(scheme, None)
#print scheme.get_str(depth = None, display_precision = True)
print "MDL instantiated scheme"
opt_eng.instantiate_precision(scheme, default_precision=ML_Binary32)
print "subexpression sharing"
opt_eng.subexpression_sharing(scheme)
print "silencing operation"
opt_eng.silence_fp_operations(scheme)
# registering scheme as function implementation
func_implementation.set_scheme(scheme)
# check processor support
opt_eng.check_processor_support(scheme)
# factorizing fast path
opt_eng.factorize_fast_path(scheme)
#print scheme.get_str(depth = None, display_precision = True)
cg = CCodeGenerator(self.processor,
declare_cst=False,
disable_debug=not debug_flag,
libm_compliant=libm_compliant)
self.result = func_implementation.get_definition(cg,
C_Code,
static_cst=True)
self.result.add_header("support_lib/ml_special_values.h")
self.result.add_header("math.h")
self.result.add_header("stdio.h")
self.result.add_header("inttypes.h")
#print self.result.get(cg)
output_stream = open("%s.c" % func_implementation.get_name(), "w")
output_stream.write(self.result.get(cg))
output_stream.close()
def generate_scheme(self):
vx = self.implementation.add_input_variable("x", self.precision)
sollya_precision = self.get_input_precision().sollya_object
# local overloading of RaiseReturn operation
def ExpRaiseReturn(*args, **kwords):
kwords["arg_value"] = vx
kwords["function_name"] = self.function_name
return RaiseReturn(*args, **kwords)
# 2-limb approximation of log(2)
# hi part precision is reduced to provide exact operation
# when multiplied by an exponent value
log2_hi_value = round(log(2), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
log2_lo_value = round(log(2) - log2_hi_value, self.precision.sollya_object, sollya.RN)
log2_hi = Constant(log2_hi_value, precision=self.precision)
log2_lo = Constant(log2_lo_value, precision=self.precision)
int_precision = self.precision.get_integer_format()
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision = self.precision)
dummy_rcp_seed = ReciprocalSeed(dummy_var, precision = self.precision)
inv_approx_table = self.processor.get_recursive_implementation(dummy_rcp_seed, language = None, table_getter = lambda self: self.approx_table_map)
# table creation
table_index_size = inv_approx_table.index_size
log_table = ML_NewTable(dimensions = [2**table_index_size, 2], storage_precision = self.precision)
# storing accurate logarithm approximation of value returned
# by the fast reciprocal operation
for i in range(0, 2**table_index_size):
inv_value = inv_approx_table[i]
value_high = round(log(inv_value), self.precision.get_field_size() - (self.precision.get_exponent_size() + 1), sollya.RN)
value_low = round(log(inv_value) - value_high, sollya_precision, sollya.RN)
log_table[i][0] = value_high
log_table[i][1] = value_low
neg_input = Comparison(vx, -1, likely=False, precision=ML_Bool, specifier=Comparison.Less, debug=debug_multi, tag="neg_input")
vx_nan_or_inf = Test(vx, specifier=Test.IsInfOrNaN, likely=False, precision=ML_Bool, debug=debug_multi, tag="nan_or_inf")
vx_snan = Test(vx, specifier=Test.IsSignalingNaN, likely=False, debug=debug_multi, tag="snan")
vx_inf = Test(vx, specifier=Test.IsInfty, likely=False, debug=debug_multi, tag="inf")
vx_subnormal = Test(vx, specifier=Test.IsSubnormal, likely=False, debug=debug_multi, tag="vx_subnormal")
# for x = m.2^e, such that e >= 0
#
# log(1+x) = log(1 + m.2^e)
# = log(2^e . 2^-e + m.2^e)
# = log(2^e . (2^-e + m))
# = log(2^e) + log(2^-e + m)
# = e . log(2) + log (2^-e + m)
#
# t = (2^-e + m)
# t = m_t . 2^e_t
# r ~ 1 / m_t => r.m_t ~ 1 ~ 0
#
# t' = t . 2^-e_t
# = 2^-e-e_t + m . 2^-e_t
#
# if e >= 0, then 2^-e <= 1, then 1 <= m + 2^-e <= 3
# r = m_r . 2^e_r
#
# log(1+x) = e.log(2) + log(r . 2^e_t . 2^-e_t . (2^-e + m) / r)
# = e.log(2) + log(r . 2^(-e-e_t) + r.m.2^-e_t) + e_t . log(2)- log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + log(r . t') - log(r)
# = (e+e_t).log(2) + P_log1p(r . t' - 1) - log(r)
#
#
# argument reduction
m = MantissaExtraction(vx, tag="vx", precision=self.precision, debug=debug_multi)
e = ExponentExtraction(vx, tag="e", precision=int_precision, debug=debug_multi)
# 2^-e
TwoMinusE = ExponentInsertion(-e, tag="Two_minus_e", precision=self.precision, debug=debug_multi)
t = Addition(TwoMinusE, m, precision=self.precision, tag="t", debug=debug_multi)
m_t = MantissaExtraction(t, tag="m_t", precision=self.precision, debug=debug_multi)
e_t = ExponentExtraction(t, tag="e_t", precision=int_precision, debug=debug_multi)
# 2^(-e-e_t)
TwoMinusEEt = ExponentInsertion(-e-e_t, tag="Two_minus_e_et", precision=self.precision)
TwoMinusEt = ExponentInsertion(-e_t, tag="Two_minus_et", precision=self.precision, debug=debug_multi)
rcp_mt = ReciprocalSeed(m_t, tag="rcp_mt", precision=self.precision, debug=debug_multi)
INDEX_SIZE = table_index_size
table_index = generic_mantissa_msb_index_fct(INDEX_SIZE, m_t)
table_index.set_attributes(tag="table_index", debug=debug_multi)
log_inv_lo = TableLoad(log_table, table_index, 1, tag="log_inv_lo", debug=debug_multi)
log_inv_hi = TableLoad(log_table, table_index, 0, tag="log_inv_hi", debug=debug_multi)
inv_err = S2**-6 # TODO: link to target DivisionSeed precision
Log.report(Log.Info, "building mathematical polynomial")
approx_interval = Interval(-inv_err, inv_err)
approx_fct = sollya.log1p(sollya.x) / (sollya.x)
poly_degree = sup(guessdegree(approx_fct, approx_interval, S2**-(self.precision.get_field_size()+1))) + 1
Log.report(Log.Debug, "poly_degree is {}", poly_degree)
global_poly_object = Polynomial.build_from_approximation(approx_fct, poly_degree, [self.precision]*(poly_degree+1), approx_interval, sollya.absolute)
poly_object = global_poly_object # .sub_poly(start_index=1)
EXT_PRECISION_MAP = {
ML_Binary32: ML_SingleSingle,
ML_Binary64: ML_DoubleDouble,
ML_SingleSingle: ML_TripleSingle,
ML_DoubleDouble: ML_TripleDouble
}
if not self.precision in EXT_PRECISION_MAP:
Log.report(Log.Error, "no extended precision available for {}", self.precision)
ext_precision = EXT_PRECISION_MAP[self.precision]
# pre_rtp = r . 2^(-e-e_t) + m .2^-e_t
pre_rtp = Addition(
rcp_mt * TwoMinusEEt,
Multiplication(
rcp_mt,
Multiplication(
m,
TwoMinusEt,
precision=self.precision,
tag="pre_mult",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_mult2",
debug=debug_multi,
),
precision=ext_precision,
tag="pre_rtp",
debug=debug_multi
)
pre_red_vx = Addition(
pre_rtp,
-1,
precision=ext_precision,
)
red_vx = Conversion(pre_red_vx, precision=self.precision, tag="red_vx", debug=debug_multi)
Log.report(Log.Info, "generating polynomial evaluation scheme")
poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object, red_vx, unified_precision=self.precision)
poly.set_attributes(tag="poly", debug=debug_multi)
Log.report(Log.Debug, "{}", global_poly_object.get_sollya_object())
fp_e = Conversion(e + e_t, precision=self.precision, tag="fp_e", debug=debug_multi)
ext_poly = Multiplication(red_vx, poly, precision=ext_precision)
pre_result = Addition(
Addition(
fp_e * log2_hi,
fp_e * log2_lo,
precision=ext_precision
),
Addition(
Addition(
-log_inv_hi,
-log_inv_lo,
precision=ext_precision
),
ext_poly,
precision=ext_precision
),
precision=ext_precision
)
result = Conversion(pre_result, precision=self.precision, tag="result", debug=debug_multi)
# main scheme
Log.report(Log.Info, "MDL scheme")
pre_scheme = ConditionBlock(neg_input,
Statement(
ClearException(),
Raise(ML_FPE_Invalid),
Return(FP_QNaN(self.precision))
),
ConditionBlock(vx_nan_or_inf,
ConditionBlock(vx_inf,
Statement(
ClearException(),
Return(FP_PlusInfty(self.precision)),
),
Statement(
ClearException(),
ConditionBlock(vx_snan,
Raise(ML_FPE_Invalid)
),
Return(FP_QNaN(self.precision))
)
),
Return(result)
)
)
scheme = pre_scheme
return scheme
def generate_scheme(self):
"""Produce an abstract scheme for the logarithm.
This abstract scheme will be used by the code generation backend.
"""
if self.precision not in [ML_Binary32, ML_Binary64]:
Log.report(Log.Error, "The demanded precision is not supported")
vx = self.implementation.add_input_variable("x", self.precision)
def default_bool_convert(optree, precision=None, **kw):
return bool_convert(optree, precision, -1, 0, **kw) \
if isinstance(self.processor, VectorBackend) \
else bool_convert(optree, precision, 1, 0, **kw)
precision = self.precision.sollya_object
int_prec = self.precision.get_integer_format()
Log.report(Log.Info, "int_prec is %s" % int_prec)
uint_prec = self.precision.get_unsigned_integer_format()
Log.report(Log.Info, "MDL constants")
cgpe_scheme_idx = int(self.cgpe_index)
table_index_size = int(self.tbl_index_size)
#
table_nb_elements = 2**(table_index_size)
table_dimensions = [2*table_nb_elements] # two values are stored for each element
field_size = Constant(self.precision.get_field_size(),
precision = int_prec,
tag = 'field_size')
if self.log_radix == EXP_1:
log2_hi = Constant(
round(log(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log(2) - round(log(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
elif self.log_radix == 10:
log2_hi = Constant(
round(log10(2), precision, sollya.RN),
precision = self.precision,
tag = 'log2_hi')
log2_lo = Constant(
round(log10(2) - round(log10(2), precision, sollya.RN),
precision, sollya.RN),
precision = self.precision,
tag = 'log2_lo')
# ... if log_radix == '2' then log2(2) == 1
# subnormal_mask aims at trapping positive subnormals except zero.
# That's why we will subtract 1 to the integer bitstring of the input, and
# then compare for Less (strict) the resulting integer bitstring to this
# mask, e.g. 0x7fffff for binary32.
if self.no_subnormal == False:
subnormal_mask = Constant((1 << self.precision.get_field_size()) - 1,
precision = int_prec, tag = 'subnormal_mask')
fp_one = Constant(1.0, precision = self.precision, tag = 'fp_one')
fp_one_as_uint = TypeCast(fp_one, precision = uint_prec,
tag = 'fp_one_as_uint')
int_zero = Constant(0, precision = int_prec, tag = 'int_zero')
int_one = Constant(1, precision = int_prec, tag = 'int_one')
table_mantissa_half_ulp = Constant(
1 << (self.precision.field_size - table_index_size - 1),
precision = int_prec
)
table_s_exp_index_mask = Constant(
~((table_mantissa_half_ulp.get_value() << 1) - 1),
precision = uint_prec
)
Log.report(Log.Info, "MDL table")
# The table holds approximations of -log(2^tau * r_i) so we first compute
# the index value for which tau changes from 1 to 0.
cut = sqrt(2.)
tau_index_limit = floor(table_nb_elements * (2./cut - 1))
sollya_logtbl = [
(-log1p(float(i) / table_nb_elements)
+ (0 if i <= tau_index_limit else log(2.))) / log(self.log_radix)
for i in range(table_nb_elements)
]
# ...
init_logtbl_hi = [
round(sollya_logtbl[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl_lo = [
round(sollya_logtbl[i] - init_logtbl_hi[i],
self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
init_logtbl = [tmp[i] for i in range(len(init_logtbl_hi)) for tmp in [init_logtbl_hi, init_logtbl_lo]]
log1p_table = ML_NewTable(dimensions = table_dimensions,
storage_precision = self.precision,
init_data = init_logtbl,
tag = 'ml_log1p_table')
# ...
if self.no_rcp:
sollya_rcptbl = [
(1/((1+float(i)/table_nb_elements)+2**(-1-int(self.tbl_index_size))))
for i in range(table_nb_elements)
]
init_rcptbl = [
round(sollya_rcptbl[i],
int(self.tbl_index_size)+1, # self.precision.get_mantissa_size(),
sollya.RN)
for i in range(table_nb_elements)
]
rcp_table = ML_NewTable(dimensions = [table_nb_elements],
storage_precision = self.precision,
init_data = init_rcptbl,
tag = 'ml_rcp_table')
# ...
Log.report(Log.Info, 'MDL unified subnormal handling')
vx_as_int = TypeCast(vx, precision = int_prec, tag = 'vx_as_int')
if self.no_subnormal == False:
vx_as_uint = TypeCast(vx, precision = uint_prec, tag = 'vx_as_uint')
# Avoid the 0.0 case by subtracting 1 from vx_as_int
tmp = Comparison(vx_as_int - 1, subnormal_mask,
specifier = Comparison.Less)
is_subnormal = default_bool_convert(
tmp, # Will catch negative values as well as NaNs with sign bit set
precision = int_prec)
is_subnormal.set_attributes(tag = "is_subnormal")
if not(isinstance(self.processor, VectorBackend)):
is_subnormal = Subtraction(Constant(0, precision = int_prec),
is_subnormal,
precision = int_prec)
#################################################
# Vectorizable integer based subnormal handling #
#################################################
# 1. lzcnt
# custom lzcount-like for subnormal numbers using FPU (see draft article)
Zi = BitLogicOr(vx_as_uint, fp_one_as_uint, precision = uint_prec, tag="Zi")
Zf = Subtraction(
TypeCast(Zi, precision = self.precision),
fp_one,
precision = self.precision,
tag="Zf")
# Zf exponent is -(nlz(x) - exponent_size).
# 2. compute shift value
# Vectorial comparison on x86+sse/avx is going to look like
# '|0x00|0xff|0x00|0x00|' and that's why we use Negate.
# But for scalar code generation, comparison will rather be either 0 or 1
# in C. Thus mask below won't be correct for a scalar implementation.
# FIXME: Can we know the backend that will be called and choose in
# consequence? Should we make something arch-agnostic instead?
#
n_value = BitLogicAnd(
Addition(
DirtyExponentExtraction(Zf, self.precision),
Constant(
self.precision.get_bias(),
precision = int_prec),
precision = int_prec),
is_subnormal,
precision = int_prec,
tag = "n_value")
alpha = Negation(n_value, tag="alpha")
#
# 3. shift left
# renormalized_mantissa = BitLogicLeftShift(vx_as_int, value)
normal_vx_as_int = BitLogicLeftShift(vx_as_int, alpha)
# 4. set exponent to the right value
# Compute the exponent to add : (p-1)-(value) + 1 = p-1-value
# The final "+ 1" comes from the fact that once renormalized, the
# floating-point datum has a biased exponent of 1
#tmp0 = Subtraction(
# field_size,
# value,
# precision = int_prec,
# tag="tmp0")
# Set the value to 0 if the number is not subnormal
#tmp1 = BitLogicAnd(tmp0, is_subnormal)
#renormalized_exponent = BitLogicLeftShift(
# tmp1,
# field_size
# )
else: # no_subnormal == True
normal_vx_as_int = vx_as_int
#normal_vx_as_int = renormalized_mantissa + renormalized_exponent
normal_vx = TypeCast(normal_vx_as_int, precision = self.precision,
tag = 'normal_vx')
# alpha = BitLogicAnd(field_size, is_subnormal, tag = 'alpha')
# XXX Extract the mantissa, see if this is supported in the x86 vector
# backend or if it still uses the support_lib.
vx_mantissa = MantissaExtraction(normal_vx, precision = self.precision)
Log.report(Log.Info, "MDL scheme")
if self.force_division == True:
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
elif self.no_rcp == False:
rcp_m = ReciprocalSeed(vx_mantissa, precision = self.precision)
if not self.processor.is_supported_operation(rcp_m):
if self.precision == ML_Binary64:
# Try using a binary32 FastReciprocal
binary32_m = Conversion(vx_mantissa, precision = ML_Binary32)
rcp_m = ReciprocalSeed(binary32_m, precision = ML_Binary32)
rcp_m = Conversion(rcp_m, precision = ML_Binary64)
if not self.processor.is_supported_operation(rcp_m):
# FIXME An approximation table could be used instead but for vector
# implementations another GATHER would be required.
# However this may well be better than a division...
rcp_m = Division(fp_one, vx_mantissa, precision = self.precision)
else: # ... use a look-up table
rcp_shift = BitLogicLeftShift(normal_vx_as_int, self.precision.get_exponent_size() + 1)
rcp_idx = BitLogicRightShift(rcp_shift, self.precision.get_exponent_size() + 1 + self.precision.get_field_size() - int(self.tbl_index_size))
rcp_m = TableLoad(rcp_table, rcp_idx, tag = 'rcp_idx',
debug = debug_multi)
#
rcp_m.set_attributes(tag = 'rcp_m')
# exponent is normally either 0 or -1, since m is in [1, 2). Possible
# optimization?
# exponent = ExponentExtraction(rcp_m, precision = self.precision,
# tag = 'exponent')
ri_round = TypeCast(
Addition(
TypeCast(rcp_m, precision = int_prec),
table_mantissa_half_ulp,
precision = int_prec
),
precision = uint_prec
)
ri_fast_rndn = BitLogicAnd(
ri_round,
table_s_exp_index_mask,
tag = 'ri_fast_rndn',
precision = uint_prec
)
# u = m * ri - 1
ul = None
if self.no_rcp == True: # ... u does not fit on a single word
tmp_u, tmp_ul = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = (self.no_fma == False))
fp_minus_one = Constant(-1.0, precision = self.precision, tag = 'fp_minus_one')
u, ul = Add212(fp_minus_one, tmp_u, tmp_ul)
u.set_attributes(tag='uh')
ul.set_attributes(tag='ul')
elif self.no_fma == False:
u = FusedMultiplyAdd(
vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fp_one,
specifier = FusedMultiplyAdd.Subtract,
tag = 'u')
else: # disable FMA
# tmph + tmpl = m * ri, where tmph ~ 1
tmph, tmpl = Mul211(vx_mantissa,
TypeCast(ri_fast_rndn, precision = self.precision),
fma = False)
# u_tmp = tmph - 1 ... exact due to Sterbenz
u_tmp = Subtraction(tmph, fp_one, precision = self.precision)
# u = u_tmp - tmpl ... exact since the result u is representable as a single word
u = Addition(u_tmp, tmpl, precision = self.precision, tag = 'u')
unneeded_bits = Constant(
self.precision.field_size - table_index_size,
precision=uint_prec,
tag="unneeded_bits"
)
assert self.precision.field_size - table_index_size >= 0
ri_bits = BitLogicRightShift(
ri_fast_rndn,
unneeded_bits,
precision = uint_prec,
tag = "ri_bits"
)
# Retrieve mantissa's MSBs + first bit of exponent, for tau computation in case
# exponent is 0 (i.e. biased 127, i.e. first bit of exponent is set.).
# In this particular case, i = 0 but tau is 1
# table_index does not need to be as long as uint_prec might be,
# try and keep it the size of size_t.
size_t_prec = ML_UInt32
signed_size_t_prec = ML_Int32
table_index_mask = Constant(
(1 << (table_index_size + 1)) - 1,
precision = size_t_prec
)
table_index = BitLogicAnd(
Conversion(ri_bits, precision = size_t_prec),
table_index_mask,
tag = 'table_index',
precision = size_t_prec
)
# Compute tau using the tau_index_limit value.
tmp = default_bool_convert(
Comparison(
TypeCast(table_index, precision = signed_size_t_prec),
Constant(tau_index_limit, precision = signed_size_t_prec),
specifier = Comparison.Greater
if isinstance(self.processor, VectorBackend)
else Comparison.LessOrEqual
),
precision = signed_size_t_prec,
tag="tmp"
)
# A true tmp will typically be -1 for VectorBackends, but 1 for standard C.
tau = Conversion(
Addition(tmp, Constant(1, precision=signed_size_t_prec), precision = signed_size_t_prec, tag="pre_add")
if isinstance(self.processor, VectorBackend)
else tmp,
precision=int_prec,
tag="pre_tau"
)
tau.set_attributes(tag = 'tau')
# Update table_index: keep only table_index_size bits
table_index_hi = BitLogicAnd(
table_index,
Constant((1 << table_index_size) - 1, precision = size_t_prec),
precision = size_t_prec
)
# table_index_hi = table_index_hi << 1
table_index_hi = BitLogicLeftShift(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_hi"
)
# table_index_lo = table_index_hi + 1
table_index_lo = Addition(
table_index_hi,
Constant(1, precision = size_t_prec),
precision = size_t_prec,
tag = "table_index_lo"
)
tbl_hi = TableLoad(log1p_table, table_index_hi, tag = 'tbl_hi',
debug = debug_multi)
tbl_lo = TableLoad(log1p_table, table_index_lo, tag = 'tbl_lo',
debug = debug_multi)
# Compute exponent e + tau - alpha, but first subtract the bias.
if self.no_subnormal == False:
tmp_eptau = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
exponent = Subtraction(tmp_eptau, alpha, precision = int_prec)
else:
exponent = Addition(
Addition(
BitLogicRightShift(
normal_vx_as_int,
field_size,
tag = 'exponent',
interval = self.precision.get_exponent_interval(),
precision = int_prec),
Constant(
self.precision.get_bias(),
precision = int_prec)),
tau,
tag = 'tmp_eptau',
precision = int_prec)
#
fp_exponent = Conversion(exponent, precision = self.precision,
tag = 'fp_exponent')
Log.report(Log.Info, 'MDL polynomial approximation')
if self.log_radix == EXP_1:
sollya_function = log(1 + sollya.x)
elif self.log_radix == 2:
sollya_function = log2(1 + sollya.x)
elif self.log_radix == 10:
sollya_function = log10(1 + sollya.x)
# ...
if self.force_division == True: # rcp accuracy is 2^(-p)
boundrcp = 2**(-self.precision.get_precision())
else:
boundrcp = 1.5 * 2**(-12) # ... see Intel intrinsics guide
if self.precision in [ML_Binary64]:
if not self.processor.is_supported_operation(rcp_m):
boundrcp = (1+boundrcp)*(1+2**(-24)) - 1
else:
boundrcp = 2**(-14) # ... see Intel intrinsics guide
arg_red_mag = boundrcp + 2**(-table_index_size-1) + boundrcp * 2**(-table_index_size-1)
if self.no_rcp == False:
approx_interval = Interval(-arg_red_mag, arg_red_mag)
else:
approx_interval = Interval(-2**(-int(self.tbl_index_size)+1),2**(-int(self.tbl_index_size)+1))
max_eps = 2**-(2*(self.precision.get_field_size()))
Log.report(Log.Info, "max acceptable error for polynomial = {}".format(float.hex(max_eps)))
poly_degree = sup(
guessdegree(
sollya_function,
approx_interval,
max_eps,
)
)
Log.report(Log.Info, "poly degree is ", poly_degree)
if self.log_radix == EXP_1:
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(2, int(poly_degree) + 1), # Force 1st 2 coeffs to 0 and 1, resp.
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree - 1),
approx_interval,
sollya.absolute,
0 + sollya._x_) # Force the first 2 coefficients to 0 and 1, resp.
else: # ... == '2' or '10'
poly_object = Polynomial.build_from_approximation(
sollya_function,
range(1, int(poly_degree) + 1), # Force 1st coeff to 0
# Emulate double-self.precision coefficient formats
[self.precision.get_mantissa_size()*2 + 1]*(poly_degree),
approx_interval,
sollya.absolute,
0) # Force the first coefficients to 0
Log.report(Log.Info, str(poly_object))
constant_precision = ML_SingleSingle if self.precision == ML_Binary32 \
else ML_DoubleDouble if self.precision == ML_Binary64 \
else None
if is_cgpe_available():
log1pu_poly = PolynomialSchemeEvaluator.generate_cgpe_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision, scheme_id = cgpe_scheme_idx
)
else:
Log.report(Log.Warning,
"CGPE not available, falling back to std poly evaluator")
log1pu_poly = PolynomialSchemeEvaluator.generate_horner_scheme(
poly_object,
u,
unified_precision = self.precision,
constant_precision = constant_precision
)
# XXX Dirty implementation of double-(self.precision) poly
def dirty_poly_node_conversion(node, variable_h, variable_l, use_fma):
return dirty_multi_node_expand(
node, self.precision, mem_map={variable_h: (variable_h, variable_l)}, fma=use_fma)
log1pu_poly_hi, log1pu_poly_lo = dirty_poly_node_conversion(log1pu_poly, u, ul,
use_fma=(self.no_fma == False))
log1pu_poly_hi.set_attributes(tag = 'log1pu_poly_hi')
log1pu_poly_lo.set_attributes(tag = 'log1pu_poly_lo')
# Compute log(2) * (e + tau - alpha)
if self.log_radix != 2: # 'e' or '10'
log2e_hi, log2e_lo = Mul212(fp_exponent, log2_hi, log2_lo,
fma = (self.no_fma == False))
# Add log1p(u)
if self.log_radix != 2: # 'e' or '10'
tmp_res_hi, tmp_res_lo = Add222(log2e_hi, log2e_lo,
log1pu_poly_hi, log1pu_poly_lo)
else:
tmp_res_hi, tmp_res_lo = Add212(fp_exponent,
log1pu_poly_hi, log1pu_poly_lo)
# Add -log(2^(tau)/m) approximation retrieved by two table lookups
logx_hi = Add122(tmp_res_hi, tmp_res_lo, tbl_hi, tbl_lo)[0]
logx_hi.set_attributes(tag = 'logx_hi')
scheme = Return(logx_hi, precision = self.precision)
return scheme
def numeric_emulate(self, input_value):
return log1p(input_value)
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