def legalize_invsqrt_seed(optree):
""" Legalize an InverseSquareRootSeed optree """
assert isinstance(optree, ReciprocalSquareRootSeed)
op_prec = optree.get_precision()
# input = 1.m_hi-m_lo * 2^e
# approx = 2^(-int(e/2)) * approx_insqrt(1.m_hi) * (e % 2 ? 1.0 : ~2**-0.5)
op_input = optree.get_input(0)
convert_back = False
approx_prec = ML_Binary32
if op_prec != approx_prec:
op_input = Conversion(op_input, precision=ML_Binary32)
convert_back = True
# TODO: fix integer precision selection
# as we are in a late code generation stage, every node's precision
# must be set
op_exp = ExponentExtraction(op_input,
tag="op_exp",
debug=debug_multi,
precision=ML_Int32)
neg_half_exp = Division(Negation(op_exp, precision=ML_Int32),
Constant(2, precision=ML_Int32),
precision=ML_Int32)
approx_exp = ExponentInsertion(neg_half_exp,
tag="approx_exp",
debug=debug_multi,
precision=approx_prec)
op_exp_parity = Modulo(op_exp,
Constant(2, precision=ML_Int32),
precision=ML_Int32)
approx_exp_correction = Select(Equal(op_exp_parity,
Constant(0, precision=ML_Int32)),
Constant(1.0, precision=approx_prec),
Select(Equal(
op_exp_parity,
Constant(-1, precision=ML_Int32)),
Constant(S2**0.5,
precision=approx_prec),
Constant(S2**-0.5,
precision=approx_prec),
precision=approx_prec),
precision=approx_prec,
tag="approx_exp_correction",
debug=debug_multi)
table_index = invsqrt_approx_table.get_index_function()(op_input)
table_index.set_attributes(tag="invsqrt_index", debug=debug_multi)
approx = Multiplication(TableLoad(invsqrt_approx_table,
table_index,
precision=approx_prec),
Multiplication(approx_exp_correction,
approx_exp,
precision=approx_prec),
tag="invsqrt_approx",
debug=debug_multi,
precision=approx_prec)
if approx_prec != op_prec:
return Conversion(approx, precision=op_prec)
else:
return approx
def subnormalize(x_list, factor, precision=None, fma=True):
""" x_list is a multi-component number with components ordered from the
most to the least siginificant.
x_list[0] must be the rounded evaluation of (x_list[0] + x_list[1] + ...)
@return the field of x as a floating-point number assuming
the exponent of the result is exponent(x) + factor
and managing field subnormalization if required """
x_hi = x_list[0]
int_precision = precision.get_integer_format()
ex = ExponentExtraction(x_hi, precision=int_precision)
scaled_ex = ex + factor
# difference betwen x's real exponent and the minimal exponent
# for a floating of format precision
CI0 = Constant(0, precision=int_precision)
CI1 = Constant(1, precision=int_precision)
delta = Max(Min(precision.get_emin() - scaled_ex, CI0),
Constant(precision.get_field_size(), precision=int_precision))
casted_int_x = TypeCast(x_hi, precision=int_precision)
# compute a constant to be added to a casted floating-point to perform
# rounding. This constant shall be equivalent to a half-ulp
round_cst = BitLogicLeftShift(CI1, delta - 1, precision=int_precision)
pre_rounded_value = TypeCast(casted_int_x + round_cst, precision=precision)
sticky_shift = precision.get_bit_size() - (delta - 1)
sticky = BitLogicLeftShift(casted_int_x,
sticky_shift,
precision=int_precision)
low_sticky_sign = CI0
if len(x_list) > 1:
for x_op in x_list[1:]:
sticky = BitLogicOr(sticky, x_op)
low_sticky_sign = BitLogicOr(
BitLogicXor(CopySign(x_hi), CopySign(x_op)), low_sticky_sign)
# does the low sticky (x_list[1:]) differs in signedness from x_hi ?
parity_bit = BitLogicAnd(casted_int_x,
BitLogicLeftShift(1,
delta,
precision=int_precision),
precision=int_precision)
inc_select = LogicalAnd(Equal(sticky, CI0), Equal(parity_bit, CI0))
rounded_value = Select(inc_select,
x,
pre_rounded_value,
precision=precision)
# cleaning trailing-bits
return TypeCast(BitLogicRightShift(BitLogicLeftShift(
TypeCast(rounded_value, precision=int_precision),
delta,
precision=int_precision),
delta,
precision=int_precision),
precision=precision)
def subnormalize_multi(x_list, factor, precision=None, fma=True):
""" x_list is a multi-component number with components ordered from the
most to the least siginificant.
x_list[0] must be the rounded evaluation of (x_list[0] + x_list[1] + ...)
@return the field of x as a floating-point number assuming
the exponent of the result is exponent(x) + factor
and managing field subnormalization if required """
x_hi = x_list[0]
int_precision = precision.get_integer_format()
ex = ExponentExtraction(x_hi, precision=int_precision)
scaled_ex = Addition(ex, factor, precision=int_precision)
CI0 = Constant(0, precision=int_precision)
CI1 = Constant(1, precision=int_precision)
# difference betwen x's real exponent and the minimal exponent
# for a floating of format precision
delta = Max(Min(Subtraction(Constant(precision.get_emin_normal(),
precision=int_precision),
scaled_ex,
precision=int_precision),
CI0,
precision=int_precision),
Constant(precision.get_field_size(), precision=int_precision),
precision=int_precision)
round_factor_exp = Addition(delta, ex, precision=int_precision)
round_factor = ExponentInsertion(round_factor_exp, precision=precision)
# to force a rounding as if x_hi was of precision p - delta
# we use round_factor as follows:
# o(o(round_factor + x_hi) - round_factor)
if len(x_list) == 2:
rounded_x_hi = Subtraction(Add112(round_factor,
x_list[0],
x_list[1],
precision=precision)[0],
round_factor,
precision=precision)
elif len(x_list) == 3:
rounded_x_hi = Subtraction(Add113(round_factor,
x_list[0],
x_list[1],
x_list[2],
precision=precision)[0],
round_factor,
precision=precision)
else:
Log.report(Log.Error,
"len of x_list: {} is not supported in subnormalize_multi",
len(x_list))
raise NotImplementedError
return [rounded_x_hi] + [
Constant(0, precision=precision) for i in range(len(x_list) - 1)
]
def legalize_reciprocal_seed(optree):
""" Legalize an ReciprocalSeed optree """
assert isinstance(optree, ReciprocalSeed)
op_prec = optree.get_precision()
initial_prec = op_prec
back_convert = False
op_input = optree.get_input(0)
INV_APPROX_TABLE_FORMAT = generic_inv_approx_table.get_storage_precision()
if op_prec != INV_APPROX_TABLE_FORMAT:
op_input = Conversion(op_input, precision=INV_APPROX_TABLE_FORMAT)
op_prec = INV_APPROX_TABLE_FORMAT
back_convert = True
# input = 1.m_hi-m_lo * 2^e
# approx = 2^(-int(e/2)) * approx_insqrt(1.m_hi) * (e % 2 ? 1.0 : ~2**-0.5)
# TODO: fix integer precision selection
# as we are in a late code generation stage, every node's precision
# must be set
int_prec = op_prec.get_integer_format()
op_sign = CopySign(op_input,
Constant(1.0, precision=op_prec),
precision=op_prec)
op_exp = ExponentExtraction(op_input,
tag="op_exp",
debug=debug_multi,
precision=int_prec)
neg_exp = Negation(op_exp, precision=int_prec)
approx_exp = ExponentInsertion(neg_exp,
tag="approx_exp",
debug=debug_multi,
precision=op_prec)
table_index = generic_inv_approx_table.get_index_function()(op_input)
table_index.set_attributes(tag="inv_index", debug=debug_multi)
approx = Multiplication(TableLoad(generic_inv_approx_table,
table_index,
precision=op_prec),
Multiplication(approx_exp,
op_sign,
precision=op_prec),
tag="inv_approx",
debug=debug_multi,
precision=op_prec)
if back_convert:
return Conversion(approx, precision=initial_prec)
else:
return approx
def mantissa_extraction_modifier(optree):
""" Legalizing a MantissaExtraction node into a sub-graph
of basic operation """
init_stage = optree.attributes.get_dyn_attribute("init_stage")
op = optree.get_input(0)
tag = optree.get_tag() or "mant_extr"
op_precision = op.get_precision().get_base_format()
exp_prec = ML_StdLogicVectorFormat(op_precision.get_exponent_size())
field_prec = ML_StdLogicVectorFormat(op_precision.get_field_size())
exp_op = ExponentExtraction(op,
precision=exp_prec,
init_stage=init_stage,
tag=tag + "_exp_extr")
field_op = SubSignalSelection(TypeCast(
op,
precision=op.get_precision().get_support_format(),
init_stage=init_stage,
tag=tag + "_field_cast"),
0,
op_precision.get_field_size() - 1,
precision=field_prec,
init_stage=init_stage,
tag=tag + "_field")
exp_is_zero = Comparison(exp_op,
Constant(op_precision.get_zero_exponent_value(),
precision=exp_prec,
init_stage=init_stage),
precision=ML_Bool,
specifier=Comparison.Equal,
init_stage=init_stage)
result = mantissa_extraction_modifier_from_fields(op, field_op,
exp_is_zero)
forward_attributes(optree, result)
return result
def generate_scalar_scheme(self, vx, vy):
# fixing inputs' node tag
vx.set_attributes(tag="x")
vy.set_attributes(tag="y")
int_precision = self.precision.get_integer_format()
# assuming x = m.2^e (m in [1, 2[)
# n, positive or null integers
#
# pow(x, n) = x^(y)
# = exp(y * log(x))
# = 2^(y * log2(x))
# = 2^(y * (log2(m) + e))
#
e = ExponentExtraction(vx, tag="e", precision=int_precision)
m = MantissaExtraction(vx, tag="m", precision=self.precision)
# approximation log2(m)
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision = self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision = self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed, language=None,
table_getter= lambda self: self.approx_table_map)
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
ml_log_args = ML_GenericLog.get_default_args(precision=self.precision, basis=2)
ml_log = ML_GenericLog(ml_log_args)
log_table, log_table_tho, table_index_range = ml_log.generate_log_table(log_f, inv_approx_table)
log_approx = ml_log.generate_reduced_log_split(Abs(m, precision=self.precision), log_f, inv_approx_table, log_table)
log_approx = Select(Equal(vx, 0), FP_MinusInfty(self.precision), log_approx)
log_approx.set_attributes(tag="log_approx", debug=debug_multi)
r = Multiplication(log_approx, vy, tag="r", debug=debug_multi)
# 2^(y * (log2(m) + e)) = 2^(y * log2(m)) * 2^(y * e)
#
# log_approx = log2(Abs(m))
# r = y * log_approx ~ y * log2(m)
#
# NOTES: manage cases where e is negative and
# (y * log2(m)) AND (y * e) could cancel out
# if e positive, whichever the sign of y (y * log2(m)) and (y * e) CANNOT
# be of opposite signs
# log2(m) in [0, 1[ so cancellation can occur only if e == -1
# we split 2^x in 2^x = 2^t0 * 2^t1
# if e < 0: t0 = y * (log2(m) + e), t1=0
# else: t0 = y * log2(m), t1 = y * e
t_cond = e < 0
# e_y ~ e * y
e_f = Conversion(e, precision=self.precision)
#t0 = Select(t_cond, (e_f + log_approx) * vy, Multiplication(e_f, vy), tag="t0")
#NearestInteger(t0, precision=self.precision, tag="t0_int")
EY = NearestInteger(e_f * vy, tag="EY", precision=self.precision)
LY = NearestInteger(log_approx * vy, tag="LY", precision=self.precision)
t0_int = Select(t_cond, EY + LY, EY, tag="t0_int")
t0_frac = Select(t_cond, FMA(e_f, vy, -EY) + FMA(log_approx, vy, -LY) ,EY - t0_int, tag="t0_frac")
#t0_frac.set_attributes(tag="t0_frac")
ml_exp2_args = ML_Exp2.get_default_args(precision=self.precision)
ml_exp2 = ML_Exp2(ml_exp2_args)
exp2_t0_frac = ml_exp2.generate_scalar_scheme(t0_frac, inline_select=True)
exp2_t0_frac.set_attributes(tag="exp2_t0_frac", debug=debug_multi)
exp2_t0_int = ExponentInsertion(Conversion(t0_int, precision=int_precision), precision=self.precision, tag="exp2_t0_int")
t1 = Select(t_cond, Constant(0, precision=self.precision), r)
exp2_t1 = ml_exp2.generate_scalar_scheme(t1, inline_select=True)
exp2_t1.set_attributes(tag="exp2_t1", debug=debug_multi)
result_sign = Constant(1.0, precision=self.precision) # Select(n_is_odd, CopySign(vx, Constant(1.0, precision=self.precision)), 1)
y_int = NearestInteger(vy, precision=self.precision)
y_is_integer = Equal(y_int, vy)
y_is_even = LogicalOr(
# if y is a number (exc. inf) greater than 2**mantissa_size * 2,
# then it is an integer multiple of 2 => even
Abs(vy) >= 2**(self.precision.get_mantissa_size()+1),
LogicalAnd(
y_is_integer and Abs(vy) < 2**(self.precision.get_mantissa_size()+1),
# we want to limit the modulo computation to an integer input
Equal(Modulo(Conversion(y_int, precision=int_precision), 2), 0)
)
)
y_is_odd = LogicalAnd(
LogicalAnd(
Abs(vy) < 2**(self.precision.get_mantissa_size()+1),
y_is_integer
),
Equal(Modulo(Conversion(y_int, precision=int_precision), 2), 1)
)
# special cases management
special_case_results = Statement(
# x is sNaN OR y is sNaN
ConditionBlock(
LogicalOr(Test(vx, specifier=Test.IsSignalingNaN), Test(vy, specifier=Test.IsSignalingNaN)),
Return(FP_QNaN(self.precision))
),
# pow(x, ±0) is 1 if x is not a signaling NaN
ConditionBlock(
Test(vy, specifier=Test.IsZero),
Return(Constant(1.0, precision=self.precision))
),
# pow(±0, y) is ±∞ and signals the divideByZero exception for y an odd integer <0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(y_is_odd, vy < 0)),
Return(Select(Test(vx, specifier=Test.IsPositiveZero), FP_PlusInfty(self.precision), FP_MinusInfty(self.precision))),
),
# pow(±0, −∞) is +∞ with no exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_MinusInfty(self.precision)),
),
# pow(±0, +∞) is +0 with no exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is ±0 for finite y>0 an odd integer
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(y_is_odd, vy > 0)),
Return(vx),
),
# pow(−1, ±∞) is 1 with no exception
ConditionBlock(
LogicalAnd(Equal(vx, -1), Test(vy, specifier=Test.IsInfty)),
Return(Constant(1.0, precision=self.precision)),
),
# pow(+1, y) is 1 for any y (even a quiet NaN)
ConditionBlock(
vx == 1,
Return(Constant(1.0, precision=self.precision)),
),
# pow(x, +∞) is +0 for −1<x<1
ConditionBlock(
LogicalAnd(Abs(vx) < 1, Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusZero(self.precision))
),
# pow(x, +∞) is +∞ for x<−1 or for 1<x (including ±∞)
ConditionBlock(
LogicalAnd(Abs(vx) > 1, Test(vy, specifier=Test.IsPositiveInfty)),
Return(FP_PlusInfty(self.precision))
),
# pow(x, −∞) is +∞ for −1<x<1
ConditionBlock(
LogicalAnd(Abs(vx) < 1, Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_PlusInfty(self.precision))
),
# pow(x, −∞) is +0 for x<−1 or for 1<x (including ±∞)
ConditionBlock(
LogicalAnd(Abs(vx) > 1, Test(vy, specifier=Test.IsNegativeInfty)),
Return(FP_PlusZero(self.precision))
),
# pow(+∞, y) is +0 for a number y < 0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsPositiveInfty), vy < 0),
Return(FP_PlusZero(self.precision))
),
# pow(+∞, y) is +∞ for a number y > 0
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsPositiveInfty), vy > 0),
Return(FP_PlusInfty(self.precision))
),
# pow(−∞, y) is −0 for finite y < 0 an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(y_is_odd, vy < 0)),
Return(FP_MinusZero(self.precision)),
),
# pow(−∞, y) is −∞ for finite y > 0 an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(y_is_odd, vy > 0)),
Return(FP_MinusInfty(self.precision)),
),
# pow(−∞, y) is +0 for finite y < 0 and not an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(LogicalNot(y_is_odd), vy < 0)),
Return(FP_PlusZero(self.precision)),
),
# pow(−∞, y) is +∞ for finite y > 0 and not an odd integer
# TODO: check y is finite
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsNegativeInfty), LogicalAnd(LogicalNot(y_is_odd), vy > 0)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is +∞ and signals the divideByZero exception for finite y<0 and not an odd integer
# TODO: signal divideByZero exception
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(LogicalNot(y_is_odd), vy < 0)),
Return(FP_PlusInfty(self.precision)),
),
# pow(±0, y) is +0 for finite y>0 and not an odd integer
ConditionBlock(
LogicalAnd(Test(vx, specifier=Test.IsZero), LogicalAnd(LogicalNot(y_is_odd), vy > 0)),
Return(FP_PlusZero(self.precision)),
),
)
# manage n=1 separately to avoid catastrophic propagation of errors
# between log2 and exp2 to eventually compute the identity function
# test-case #3
result = Statement(
special_case_results,
# fallback default cases
Return(result_sign * exp2_t1 * exp2_t0_int * exp2_t0_frac))
return result
def generate_scheme(self):
# We wish to compute vx / vy
vx = self.implementation.add_input_variable(
"x", self.precision, interval=self.input_intervals[0])
vy = self.implementation.add_input_variable(
"y", self.precision, interval=self.input_intervals[1])
# maximum exponent magnitude (to avoid overflow/ underflow during
# intermediary computations
int_prec = self.precision.get_integer_format()
max_exp_mag = Constant(self.precision.get_emax() - 1,
precision=int_prec)
exact_ex = ExponentExtraction(vx,
tag="exact_ex",
precision=int_prec,
debug=debug_multi)
exact_ey = ExponentExtraction(vy,
tag="exact_ey",
precision=int_prec,
debug=debug_multi)
ex = Max(Min(exact_ex, max_exp_mag, precision=int_prec),
-max_exp_mag,
tag="ex",
precision=int_prec)
ey = Max(Min(exact_ey, max_exp_mag, precision=int_prec),
-max_exp_mag,
tag="ey",
precision=int_prec)
Attributes.set_default_rounding_mode(ML_RoundToNearest)
Attributes.set_default_silent(True)
# computing the inverse square root
init_approx = None
scaling_factor_x = ExponentInsertion(-ex,
tag="sfx_ei",
precision=self.precision,
debug=debug_multi)
scaling_factor_y = ExponentInsertion(-ey,
tag="sfy_ei",
precision=self.precision,
debug=debug_multi)
def test_interval_out_of_bound_risk(x_range, y_range):
""" Try to determine from x and y's interval if there is a risk
of underflow or overflow """
div_range = abs(x_range / y_range)
underflow_risk = sollya.inf(div_range) < S2**(
self.precision.get_emin_normal() + 2)
overflow_risk = sollya.sup(div_range) > S2**(
self.precision.get_emax() - 2)
return underflow_risk or overflow_risk
out_of_bound_risk = (self.input_intervals[0] is None
or self.input_intervals[1] is None
) or test_interval_out_of_bound_risk(
self.input_intervals[0],
self.input_intervals[1])
Log.report(Log.Debug,
"out_of_bound_risk: {}".format(out_of_bound_risk))
# scaled version of vx and vy, to avoid overflow and underflow
if out_of_bound_risk:
scaled_vx = vx * scaling_factor_x
scaled_vy = vy * scaling_factor_y
scaled_interval = MetaIntervalList(
[MetaInterval(Interval(-2, -1)),
MetaInterval(Interval(1, 2))])
scaled_vx.set_attributes(tag="scaled_vx",
debug=debug_multi,
interval=scaled_interval)
scaled_vy.set_attributes(tag="scaled_vy",
debug=debug_multi,
interval=scaled_interval)
seed_interval = 1 / scaled_interval
print("seed_interval=1/{}={}".format(scaled_interval,
seed_interval))
else:
scaled_vx = vx
scaled_vy = vy
seed_interval = 1 / scaled_vy.get_interval()
# We need a first approximation to 1 / scaled_vy
dummy_seed = ReciprocalSeed(EmptyOperand(precision=self.precision),
precision=self.precision)
if self.processor.is_supported_operation(dummy_seed, self.language):
init_approx = ReciprocalSeed(scaled_vy,
precision=self.precision,
tag="init_approx",
debug=debug_multi)
else:
# generate tabulated version of seed
raise NotImplementedError
current_approx_std = init_approx
# correctly-rounded inverse computation
num_iteration = self.num_iter
Attributes.unset_default_rounding_mode()
Attributes.unset_default_silent()
# check if inputs are zeros
x_zero = Test(vx,
specifier=Test.IsZero,
likely=False,
precision=ML_Bool)
y_zero = Test(vy,
specifier=Test.IsZero,
likely=False,
precision=ML_Bool)
comp_sign = Test(vx,
vy,
specifier=Test.CompSign,
tag="comp_sign",
debug=debug_multi)
# check if divisor is NaN
y_nan = Test(vy, specifier=Test.IsNaN, likely=False, precision=ML_Bool)
# check if inputs are signaling NaNs
x_snan = Test(vx,
specifier=Test.IsSignalingNaN,
likely=False,
precision=ML_Bool)
y_snan = Test(vy,
specifier=Test.IsSignalingNaN,
likely=False,
precision=ML_Bool)
# check if inputs are infinities
x_inf = Test(vx,
specifier=Test.IsInfty,
likely=False,
tag="x_inf",
precision=ML_Bool)
y_inf = Test(vy,
specifier=Test.IsInfty,
likely=False,
tag="y_inf",
debug=debug_multi,
precision=ML_Bool)
scheme = None
gappa_vx, gappa_vy = None, None
# initial reciprocal approximation of 1.0 / scaled_vy
inv_iteration_list, recp_approx = compute_reduced_reciprocal(
init_approx, scaled_vy, self.num_iter)
recp_approx.set_attributes(tag="recp_approx", debug=debug_multi)
# approximation of scaled_vx / scaled_vy
yerr_last, reduced_div_approx, div_iteration_list = compute_reduced_division(
scaled_vx, scaled_vy, recp_approx)
eval_error_range, div_eval_error_range = self.solve_eval_error(
init_approx, recp_approx, reduced_div_approx, scaled_vx, scaled_vy,
inv_iteration_list, div_iteration_list, S2**-7, seed_interval)
eval_error = sup(abs(eval_error_range))
recp_interval = 1 / scaled_vy.get_interval() + eval_error_range
recp_approx.set_interval(recp_interval)
div_interval = scaled_vx.get_interval() / scaled_vy.get_interval(
) + div_eval_error_range
reduced_div_approx.set_interval(div_interval)
reduced_div_approx.set_tag("reduced_div_approx")
if out_of_bound_risk:
unscaled_result = scaling_div_result(reduced_div_approx, ex,
scaling_factor_y,
self.precision)
subnormal_result = subnormalize_result(recp_approx,
reduced_div_approx, ex, ey,
yerr_last, self.precision)
else:
unscaled_result = reduced_div_approx
subnormal_result = reduced_div_approx
x_inf_or_nan = Test(vx, specifier=Test.IsInfOrNaN, likely=False)
y_inf_or_nan = Test(vy,
specifier=Test.IsInfOrNaN,
likely=False,
tag="y_inf_or_nan",
debug=debug_multi)
# generate IEEE exception raising only of libm-compliant
# mode is enabled
enable_raise = self.libm_compliant
# managing special cases
# x inf and y inf
pre_scheme = ConditionBlock(
x_inf_or_nan,
ConditionBlock(
x_inf,
ConditionBlock(
y_inf_or_nan,
Statement(
# signaling NaNs raise invalid operation flags
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)),
),
ConditionBlock(comp_sign,
Return(FP_MinusInfty(self.precision)),
Return(FP_PlusInfty(self.precision)))),
Statement(
ConditionBlock(x_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
x_zero,
ConditionBlock(
LogicalOr(y_zero, y_nan, precision=ML_Bool),
Statement(
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision))), Return(vx)),
ConditionBlock(
y_inf_or_nan,
ConditionBlock(
y_inf,
Return(
Select(comp_sign, FP_MinusZero(self.precision),
FP_PlusZero(self.precision))),
Statement(
ConditionBlock(y_snan, Raise(ML_FPE_Invalid))
if enable_raise else Statement(),
Return(FP_QNaN(self.precision)))),
ConditionBlock(
y_zero,
Statement(
Raise(ML_FPE_DivideByZero)
if enable_raise else Statement(),
ConditionBlock(
comp_sign,
Return(FP_MinusInfty(self.precision)),
Return(FP_PlusInfty(self.precision)))),
# managing numerical value result cases
Statement(
recp_approx,
reduced_div_approx,
ConditionBlock(
Test(unscaled_result,
specifier=Test.IsSubnormal,
likely=False),
# result is subnormal
Statement(
# inexact flag should have been raised when computing yerr_last
# ConditionBlock(
# Comparison(
# yerr_last, 0,
# specifier=Comparison.NotEqual, likely=True),
# Statement(Raise(ML_FPE_Inexact, ML_FPE_Underflow))
#),
Return(subnormal_result), ),
# result is normal
Statement(
# inexact flag should have been raised when computing yerr_last
#ConditionBlock(
# Comparison(
# yerr_last, 0,
# specifier=Comparison.NotEqual, likely=True),
# Raise(ML_FPE_Inexact)
#),
Return(unscaled_result))),
)))))
# managing rounding mode save and restore
# to ensure intermediary computations are performed in round-to-nearest
# clearing exception before final computation
#rnd_mode = GetRndMode()
#scheme = Statement(
# rnd_mode,
# SetRndMode(ML_RoundToNearest),
# yerr_last,
# SetRndMode(rnd_mode),
# unscaled_result,
# ClearException(),
# pre_scheme
#)
scheme = pre_scheme
return scheme
def generate_scalar_scheme(self, vx, n):
# fixing inputs' node tag
vx.set_attributes(tag="x")
n.set_attributes(tag="n")
int_precision = self.precision.get_integer_format()
# assuming x = m.2^e (m in [1, 2[)
# n, positive or null integers
#
# rootn(x, n) = x^(1/n)
# = exp(1/n * log(x))
# = 2^(1/n * log2(x))
# = 2^(1/n * (log2(m) + e))
#
# approximation log2(m)
# retrieving processor inverse approximation table
dummy_var = Variable("dummy", precision=self.precision)
dummy_div_seed = ReciprocalSeed(dummy_var, precision=self.precision)
inv_approx_table = self.processor.get_recursive_implementation(
dummy_div_seed,
language=None,
table_getter=lambda self: self.approx_table_map)
log_f = sollya.log(sollya.x) # /sollya.log(self.basis)
use_reciprocal = False
# non-scaled vx used to compute vx^1
unmodified_vx = vx
is_subnormal = Test(vx, specifier=Test.IsSubnormal, tag="is_subnormal")
exp_correction_factor = self.precision.get_mantissa_size()
mantissa_factor = Constant(2**exp_correction_factor,
tag="mantissa_factor")
vx = Select(is_subnormal, vx * mantissa_factor, vx, tag="corrected_vx")
m = MantissaExtraction(vx, tag="m", precision=self.precision)
e = ExponentExtraction(vx, tag="e", precision=int_precision)
e = Select(is_subnormal,
e - exp_correction_factor,
e,
tag="corrected_e")
ml_log_args = ML_GenericLog.get_default_args(precision=self.precision,
basis=2)
ml_log = ML_GenericLog(ml_log_args)
log_table, log_table_tho, table_index_range = ml_log.generate_log_table(
log_f, inv_approx_table)
log_approx = ml_log.generate_reduced_log_split(
Abs(m, precision=self.precision), log_f, inv_approx_table,
log_table)
# floating-point version of n
n_f = Conversion(n, precision=self.precision, tag="n_f")
inv_n = Division(Constant(1, precision=self.precision), n_f)
log_approx = Select(Equal(vx, 0), FP_MinusInfty(self.precision),
log_approx)
log_approx.set_attributes(tag="log_approx", debug=debug_multi)
if use_reciprocal:
r = Multiplication(log_approx, inv_n, tag="r", debug=debug_multi)
else:
r = Division(log_approx, n_f, tag="r", debug=debug_multi)
# e_n ~ e / n
e_f = Conversion(e, precision=self.precision, tag="e_f")
if use_reciprocal:
e_n = Multiplication(e_f, inv_n, tag="e_n")
else:
e_n = Division(e_f, n_f, tag="e_n")
error_e_n = FMA(e_n, -n_f, e_f, tag="error_e_n")
e_n_int = NearestInteger(e_n, precision=self.precision, tag="e_n_int")
pre_e_n_frac = e_n - e_n_int
pre_e_n_frac.set_attributes(tag="pre_e_n_frac")
e_n_frac = pre_e_n_frac + error_e_n * inv_n
e_n_frac.set_attributes(tag="e_n_frac")
ml_exp2_args = ML_Exp2.get_default_args(precision=self.precision)
ml_exp2 = ML_Exp2(ml_exp2_args)
exp2_r = ml_exp2.generate_scalar_scheme(r, inline_select=True)
exp2_r.set_attributes(tag="exp2_r", debug=debug_multi)
exp2_e_n_frac = ml_exp2.generate_scalar_scheme(e_n_frac,
inline_select=True)
exp2_e_n_frac.set_attributes(tag="exp2_e_n_frac", debug=debug_multi)
exp2_e_n_int = ExponentInsertion(Conversion(e_n_int,
precision=int_precision),
precision=self.precision,
tag="exp2_e_n_int")
n_is_even = Equal(Modulo(n, 2), 0, tag="n_is_even", debug=debug_multi)
n_is_odd = LogicalNot(n_is_even, tag="n_is_odd")
result_sign = Select(
n_is_odd, CopySign(vx, Constant(1.0, precision=self.precision)), 1)
# managing n == -1
if self.expand_div:
ml_division_args = ML_Division.get_default_args(
precision=self.precision, input_formats=[self.precision] * 2)
ml_division = ML_Division(ml_division_args)
self.division_implementation = ml_division.implementation
self.division_implementation.set_scheme(
ml_division.generate_scheme())
ml_division_fct = self.division_implementation.get_function_object(
)
else:
ml_division_fct = Division
# manage n=1 separately to avoid catastrophic propagation of errors
# between log2 and exp2 to eventually compute the identity function
# test-case #3
result = ConditionBlock(
LogicalOr(LogicalOr(Test(vx, specifier=Test.IsNaN), Equal(n, 0)),
LogicalAnd(n_is_even, vx < 0)),
Return(FP_QNaN(self.precision)),
Statement(
ConditionBlock(
Equal(n, -1, tag="n_is_mone"),
#Return(Division(Constant(1, precision=self.precision), unmodified_vx, tag="div_res", precision=self.precision)),
Return(
ml_division_fct(Constant(1, precision=self.precision),
unmodified_vx,
tag="div_res",
precision=self.precision)),
),
ConditionBlock(
# rootn( ±inf, n) is +∞ for even n< 0.
Test(vx, specifier=Test.IsInfty),
Statement(
ConditionBlock(
n < 0,
#LogicalAnd(n_is_odd, n < 0),
Return(
Select(Test(vx,
specifier=Test.IsPositiveInfty),
Constant(FP_PlusZero(self.precision),
precision=self.precision),
Constant(FP_MinusZero(self.precision),
precision=self.precision),
precision=self.precision)),
Return(vx),
), ),
),
ConditionBlock(
# rootn(±0, n) is ±∞ for odd n < 0.
LogicalAnd(LogicalAnd(n_is_odd, n < 0),
Equal(vx, 0),
tag="n_is_odd_and_neg"),
Return(
Select(Test(vx, specifier=Test.IsPositiveZero),
Constant(FP_PlusInfty(self.precision),
precision=self.precision),
Constant(FP_MinusInfty(self.precision),
precision=self.precision),
precision=self.precision)),
),
ConditionBlock(
# rootn( ±0, n) is +∞ for even n< 0.
LogicalAnd(LogicalAnd(n_is_even, n < 0), Equal(vx, 0)),
Return(FP_PlusInfty(self.precision))),
ConditionBlock(
# rootn(±0, n) is +0 for even n > 0.
LogicalAnd(n_is_even, Equal(vx, 0)),
Return(vx)),
ConditionBlock(
Equal(n, 1), Return(unmodified_vx),
Return(result_sign * exp2_r * exp2_e_n_int *
exp2_e_n_frac))))
return result
def generate_scheme(self):
""" main scheme generation """
int_size = 3
frac_size = self.width - int_size
input_precision = fixed_point(int_size, frac_size)
output_precision = fixed_point(int_size, frac_size)
expected_interval = {}
# declaring main input variable
var_x = self.implementation.add_input_signal("x", input_precision)
x_interval = Interval(-10.3,10.7)
var_x.set_interval(x_interval)
expected_interval[var_x] = x_interval
var_y = self.implementation.add_input_signal("y", input_precision)
y_interval = Interval(-17.9,17.2)
var_y.set_interval(y_interval)
expected_interval[var_y] = y_interval
var_z = self.implementation.add_input_signal("z", input_precision)
z_interval = Interval(-7.3,7.7)
var_z.set_interval(z_interval)
expected_interval[var_z] = z_interval
cst = Constant(42.5, tag = "cst")
expected_interval[cst] = Interval(42.5)
conv_ceil = Ceil(var_x, tag = "ceil")
expected_interval[conv_ceil] = sollya.ceil(x_interval)
conv_floor = Floor(var_y, tag = "floor")
expected_interval[conv_floor] = sollya.floor(y_interval)
mult = var_z * var_x
mult.set_tag("mult")
mult_interval = z_interval * x_interval
expected_interval[mult] = mult_interval
large_add = (var_x + var_y) - mult
large_add.set_attributes(tag = "large_add")
large_add_interval = (x_interval + y_interval) - mult_interval
expected_interval[large_add] = large_add_interval
var_x_lzc = CountLeadingZeros(var_x, tag="var_x_lzc")
expected_interval[var_x_lzc] = Interval(0, input_precision.get_bit_size())
reduced_result = Max(0, Min(large_add, 13))
reduced_result.set_tag("reduced_result")
reduced_result_interval = interval_max(
Interval(0),
interval_min(
large_add_interval,
Interval(13)
)
)
expected_interval[reduced_result] = reduced_result_interval
select_result = Select(
var_x > var_y,
reduced_result,
var_z,
tag = "select_result"
)
select_interval = interval_union(reduced_result_interval, z_interval)
expected_interval[select_result] = select_interval
# floating-point operation on mantissa and exponents
fp_x_range = Interval(-0.01, 100)
unbound_fp_var = Variable("fp_x", precision=ML_Binary32, interval=fp_x_range)
mant_fp_x = MantissaExtraction(unbound_fp_var, tag="mant_fp_x", precision=ML_Binary32)
exp_fp_x = ExponentExtraction(unbound_fp_var, tag="exp_fp_x", precision=ML_Int32)
ins_exp_fp_x = ExponentInsertion(exp_fp_x, tag="ins_exp_fp_x", precision=ML_Binary32)
expected_interval[unbound_fp_var] = fp_x_range
expected_interval[exp_fp_x] = Interval(
sollya.floor(sollya.log2(sollya.inf(abs(fp_x_range)))),
sollya.floor(sollya.log2(sollya.sup(abs(fp_x_range))))
)
expected_interval[mant_fp_x] = Interval(1, 2)
expected_interval[ins_exp_fp_x] = Interval(
S2**sollya.inf(expected_interval[exp_fp_x]),
S2**sollya.sup(expected_interval[exp_fp_x])
)
# checking interval evaluation
for var in [var_x_lzc, exp_fp_x, unbound_fp_var, mant_fp_x, ins_exp_fp_x, cst, var_x, var_y, mult, large_add, reduced_result, select_result, conv_ceil, conv_floor]:
interval = evaluate_range(var)
expected = expected_interval[var]
print("{}: {}".format(var.get_tag(), interval))
print(" vs expected {}".format(expected))
assert not interval is None
assert interval == expected
return [self.implementation]
def generate_scheme(self):
int_precision = self.precision.get_integer_format()
# We wish to compute vx / vy
vx = self.implementation.add_input_variable("x", self.precision, interval=self.input_intervals[0])
vy = self.implementation.add_input_variable("y", self.precision, interval=self.input_intervals[1])
if self.mode is FULL_MODE:
quo = self.implementation.add_input_variable("quo", ML_Pointer_Format(int_precision))
i = Variable("i", precision=int_precision, var_type=Variable.Local)
q = Variable("q", precision=int_precision, var_type=Variable.Local)
CI = lambda v: Constant(v, precision=int_precision)
CF = lambda v: Constant(v, precision=self.precision)
vx_subnormal = Test(vx, specifier=Test.IsSubnormal, tag="vx_subnormal")
vy_subnormal = Test(vy, specifier=Test.IsSubnormal, tag="vy_subnormal")
DELTA_EXP = self.precision.get_mantissa_size()
scale_factor = Constant(2.0**DELTA_EXP, precision=self.precision)
inv_scale_factor = Constant(2.0**-DELTA_EXP, precision=self.precision)
normalized_vx = Select(vx_subnormal, vx * scale_factor, vx, tag="scaled_vx")
normalized_vy = Select(vy_subnormal, vy * scale_factor, vy, tag="scaled_vy")
real_ex = ExponentExtraction(vx, tag="real_ex", precision=int_precision)
real_ey = ExponentExtraction(vy, tag="real_ey", precision=int_precision)
# if real_e<x/y> is +1023 then it may Overflow in -real_ex for ExponentInsertion
# which only supports downto -1022 before falling into subnormal numbers (which are
# not supported by ExponentInsertion)
real_ex_h0 = real_ex / 2
real_ex_h1 = real_ex - real_ex_h0
real_ey_h0 = real_ey / 2
real_ey_h1 = real_ey - real_ey_h0
EI = lambda v: ExponentInsertion(v, precision=self.precision)
mx = Abs((vx * EI(-real_ex_h0)) * EI(-real_ex_h1), tag="mx")
my = Abs((vy * EI(-real_ey_h0)) * EI(-real_ey_h1), tag="pre_my")
# scale_ey is used to regain the unscaling of mx in the first loop
# if real_ey >= real_ex, the first loop is never executed
# so a different scaling is required
mx_unscaling = Select(real_ey < real_ex, real_ey, real_ex)
ey_half0 = (mx_unscaling) / 2
ey_half1 = (mx_unscaling) - ey_half0
scale_ey_half0 = ExponentInsertion(ey_half0, precision=self.precision, tag="scale_ey_half0")
scale_ey_half1 = ExponentInsertion(ey_half1, precision=self.precision, tag="scale_ey_half1")
# if only vy is subnormal we want to normalize it
#normal_cond = LogicalAnd(vy_subnormal, LogicalNot(vx_subnormal))
normal_cond = vy_subnormal #LogicalAnd(vy_subnormal, LogicalNot(vx_subnormal))
my = Select(normal_cond, Abs(MantissaExtraction(vy * scale_factor)), my, tag="my")
# vx / vy = vx * 2^-ex * 2^(ex-ey) / (vy * 2^-ey)
# vx % vy
post_mx = Variable("post_mx", precision=self.precision, var_type=Variable.Local)
# scaling for half comparison
VY_SCALING = Select(vy_subnormal, 1.0, 0.5, precision=self.precision)
VX_SCALING = Select(vy_subnormal, 2.0, 1.0, precision=self.precision)
def LogicalXor(a, b):
return LogicalOr(LogicalAnd(a, LogicalNot(b)), LogicalAnd(LogicalNot(a), b))
rem_sign = Select(vx < 0, CF(-1), CF(1), precision=self.precision, tag="rem_sign")
quo_sign = Select(LogicalXor(vx <0, vy < 0), CI(-1), CI(1), precision=int_precision, tag="quo_sign")
loop_watchdog = Variable("loop_watchdog", precision=ML_Int32, var_type=Variable.Local)
loop = Statement(
real_ex, real_ey, mx, my, loop_watchdog,
ReferenceAssign(loop_watchdog, 5000),
ReferenceAssign(q, CI(0)),
Loop(
ReferenceAssign(i, CI(0)), i < (real_ex - real_ey),
Statement(
ReferenceAssign(i, i+CI(1)),
ReferenceAssign(q, ((q << 1) + Select(mx >= my, CI(1), CI(0))).modify_attributes(tag="step1_q")),
ReferenceAssign(mx, (CF(2) * (mx - Select(mx >= my, my, CF(0)))).modify_attributes(tag="step1_mx")),
# loop watchdog
ReferenceAssign(loop_watchdog, loop_watchdog - 1),
ConditionBlock(loop_watchdog < 0, Return(-1)),
),
),
# unscaling remainder
ReferenceAssign(mx, ((mx * scale_ey_half0) * scale_ey_half1).modify_attributes(tag="scaled_rem")),
ReferenceAssign(my, ((my * scale_ey_half0) * scale_ey_half1).modify_attributes(tag="scaled_rem_my")),
Loop(
Statement(), (my > Abs(vy)),
Statement(
ReferenceAssign(q, ((q << 1) + Select(mx >= Abs(my), CI(1), CI(0))).modify_attributes(tag="step2_q")),
ReferenceAssign(mx, (mx - Select(mx >= Abs(my), Abs(my), CF(0))).modify_attributes(tag="step2_mx")),
ReferenceAssign(my, (my * 0.5).modify_attributes(tag="step2_my")),
# loop watchdog
ReferenceAssign(loop_watchdog, loop_watchdog - 1),
ConditionBlock(loop_watchdog < 0, Return(-1)),
),
),
ReferenceAssign(q, q << 1),
Loop(
ReferenceAssign(i, CI(0)), mx > Abs(vy),
Statement(
ReferenceAssign(q, (q + Select(mx > Abs(vy), CI(1), CI(0))).modify_attributes(tag="step3_q")),
ReferenceAssign(mx, (mx - Select(mx > Abs(vy), Abs(vy), CF(0))).modify_attributes(tag="step3_mx")),
# loop watchdog
ReferenceAssign(loop_watchdog, loop_watchdog - 1),
ConditionBlock(loop_watchdog < 0, Return(-1)),
),
),
ReferenceAssign(q, q + Select(mx >= Abs(vy), CI(1), CI(0))),
ReferenceAssign(mx, (mx - Select(mx >= Abs(vy), Abs(vy), CF(0))).modify_attributes(tag="pre_half_mx")),
ConditionBlock(
# actual comparison is mx > | abs(vy * 0.5) | to avoid rounding effect when
# vy is subnormal we mulitply both side by 2.0**60
((mx * VX_SCALING) > Abs(vy * VY_SCALING)).modify_attributes(tag="half_test"),
Statement(
ReferenceAssign(q, q + CI(1)),
ReferenceAssign(mx, (mx - Abs(vy)))
)
),
ConditionBlock(
# if the remainder is exactly half the dividend
# we need to make sure the quotient is even
LogicalAnd(
Equal(mx * VX_SCALING, Abs(vy * VY_SCALING)),
Equal(Modulo(q, CI(2)), CI(1)),
),
Statement(
ReferenceAssign(q, q + CI(1)),
ReferenceAssign(mx, (mx - Abs(vy)))
)
),
ReferenceAssign(mx, rem_sign * mx),
ReferenceAssign(q,
Modulo(TypeCast(q, precision=self.precision.get_unsigned_integer_format()), Constant(2**self.quotient_size, precision=self.precision.get_unsigned_integer_format()), tag="mod_q")
),
ReferenceAssign(q, quo_sign * q),
)
# NOTES: Warning QuotientReturn must always preceeds RemainderReturn
if self.mode is QUOTIENT_MODE:
#
QuotientReturn = Return
RemainderReturn = lambda _: Statement()
elif self.mode is REMAINDER_MODE:
QuotientReturn = lambda _: Statement()
RemainderReturn = Return
elif self.mode is FULL_MODE:
QuotientReturn = lambda v: ReferenceAssign(Dereference(quo, precision=int_precision), v)
RemainderReturn = Return
else:
raise NotImplemented
# quotient invalid value
QUO_INVALID_VALUE = 0
mod_scheme = Statement(
# x or y is NaN, a NaN is returned
ConditionBlock(
LogicalOr(Test(vx, specifier=Test.IsNaN), Test(vy, specifier=Test.IsNaN)),
Statement(
QuotientReturn(QUO_INVALID_VALUE),
RemainderReturn(FP_QNaN(self.precision))
),
),
#
ConditionBlock(
Test(vy, specifier=Test.IsZero),
Statement(
QuotientReturn(QUO_INVALID_VALUE),
RemainderReturn(FP_QNaN(self.precision))
),
),
ConditionBlock(
Test(vx, specifier=Test.IsZero),
Statement(
QuotientReturn(0),
RemainderReturn(vx)
),
),
ConditionBlock(
Test(vx, specifier=Test.IsInfty),
Statement(
QuotientReturn(QUO_INVALID_VALUE),
RemainderReturn(FP_QNaN(self.precision))
)
),
ConditionBlock(
Test(vy, specifier=Test.IsInfty),
Statement(
QuotientReturn(0),
RemainderReturn(vx),
)
),
ConditionBlock(
Abs(vx) < Abs(vy * 0.5),
Statement(
QuotientReturn(0),
RemainderReturn(vx),
)
),
ConditionBlock(
Equal(vx, vy),
Statement(
QuotientReturn(1),
# 0 with the same sign as x
RemainderReturn(vx - vx),
),
),
ConditionBlock(
Equal(vx, -vy),
Statement(
# quotient is -1
QuotientReturn(-1),
# 0 with the same sign as x
RemainderReturn(vx - vx),
),
),
loop,
QuotientReturn(q),
RemainderReturn(mx),
)
quo_scheme = Statement(
# x or y is NaN, a NaN is returned
ConditionBlock(
LogicalOr(Test(vx, specifier=Test.IsNaN), Test(vy, specifier=Test.IsNaN)),
Return(QUO_INVALID_VALUE),
),
#
ConditionBlock(
Test(vy, specifier=Test.IsZero),
Return(QUO_INVALID_VALUE),
),
ConditionBlock(
Test(vx, specifier=Test.IsZero),
Return(0),
),
ConditionBlock(
Test(vx, specifier=Test.IsInfty),
Return(QUO_INVALID_VALUE),
),
ConditionBlock(
Test(vy, specifier=Test.IsInfty),
Return(QUO_INVALID_VALUE),
),
ConditionBlock(
Abs(vx) < Abs(vy * 0.5),
Return(0),
),
ConditionBlock(
Equal(vx, vy),
Return(1),
),
ConditionBlock(
Equal(vx, -vy),
Return(-1),
),
loop,
Return(q),
)
return mod_scheme