def legalize_mp_3elt_comparison(optree): """ Transform comparison on ML_Compound_FP_Format object into comparison on sub-fields """ specifier = optree.specifier lhs = optree.get_input(0) rhs = optree.get_input(1) # TODO/FIXME: assume than multi-limb operand are normalized if specifier == Comparison.Equal: return LogicalAnd( Comparison(lhs.hi, rhs.hi, specifier=Comparison.Equal, precision=ML_Bool), LogicalAnd( Comparison(lhs.me, rhs.me, specifier=Comparison.Equal, precision=ML_Bool), Comparison(lhs.lo, rhs.lo, specifier=Comparison.Equal, precision=ML_Bool), precision=ML_Bool ), precision=ML_Bool ) elif specifier == Comparison.NotEqual: return LogicalOr( Comparison(lhs.hi, rhs.hi, specifier=Comparison.NotEqual, precision=ML_Bool), LogicalOr( Comparison(lhs.me, rhs.me, specifier=Comparison.NotEqual, precision=ML_Bool), Comparison(lhs.lo, rhs.lo, specifier=Comparison.NotEqual, precision=ML_Bool), precision=ML_Bool ), precision=ML_Bool ) elif specifier in [Comparison.LessOrEqual, Comparison.GreaterOrEqual, Comparison.Greater, Comparison.Less]: strict_specifier = { Comparison.Less: Comparison.Less, Comparison.Greater: Comparison.Greater, Comparison.LessOrEqual: Comparison.Less, Comparison.GreaterOrEqual: Comparison.Greater }[specifier] return LogicalOr( Comparison(lhs.hi, rhs.hi, specifier=strict_specifier, precision=ML_Bool), LogicalAnd( Comparison(lhs.hi, rhs.hi, specifier=Comparison.Equal, precision=ML_Bool), LogicalOr( Comparison(lhs.me, rhs.me, specifier=strict_specifier, precision=ML_Bool), LogicalAnd( Comparison(lhs.me, rhs.me, specifier=Comparison.Equal, precision=ML_Bool), Comparison(lhs.lo, rhs.lo, specifier=specifier, precision=ML_Bool), precision=ML_Bool ), precision=ML_Bool ), precision=ML_Bool ), precision=ML_Bool ) else: Log.report(Log.Error, "unsupported specifier {} in legalize_mp_2elt_comparison", specifier)
def legalize_comp_sign(node): """ legalize a Test.CompSign node to a series of comparison with 0 and logical operation """ # TODO/IDEA: could also be implemented by two 2 copy sign with 1.0 and valuda # comparison lhs = node.get_input(0) lhs_zero = Constant(0, precision=lhs.get_precision()) rhs = node.get_input(1) rhs_zero = Constant(0, precision=rhs.get_precision()) return LogicalOr( LogicalAnd(lhs >= lhs_zero, rhs >= rhs_zero), LogicalAnd(lhs <= lhs_zero, rhs <= rhs_zero), )
def generate_scheme(self): """ main scheme generation """ input_precision = self.precision output_precision = self.precision # declaring main input variable x_interval = Interval(-10.3, 10.7) var_x = self.implementation.add_input_variable("x", input_precision, interval=x_interval) y_interval = Interval(-17.9, 17.2) var_y = self.implementation.add_input_variable("y", input_precision, interval=y_interval) z_interval = Interval(-70.3, -57.7) var_z = self.implementation.add_input_variable("z", input_precision, interval=z_interval) min_yz = Min(var_z, var_y) cst0 = Constant(42.5, tag="cst0", precision=self.precision) cst1 = Constant(2.5, tag="cst1", precision=self.precision) cst2 = Constant(12.5, tag="cst2", precision=self.precision) new_cst = cst0 + cst1 * cst2 result = min_yz + new_cst scheme = ConditionBlock( LogicalAnd( LogicalOr(cst0 > cst1, LogicalNot(cst1 > cst0)), var_x > var_y, ), Return(result), Return(cst2)) return scheme
def generate_scalar_scheme(self, vx): Log.set_dump_stdout(True) Log.report(Log.Info, "\033[33;1m generating implementation scheme \033[0m") if self.debug_flag: Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m") # local overloading of RaiseReturn operation def ExpRaiseReturn(*args, **kwords): kwords["arg_value"] = vx kwords["function_name"] = self.function_name if self.libm_compliant: return RaiseReturn(*args, precision=self.precision, **kwords) else: return Return(kwords["return_value"], precision=self.precision) test_nan_or_inf = Test( vx, specifier=Test.IsInfOrNaN, likely=False, debug=debug_multi, tag="nan_or_inf") test_nan = Test( vx, specifier=Test.IsNaN, debug=debug_multi, tag="is_nan_test") test_positive = Comparison( vx, 0, specifier=Comparison.GreaterOrEqual, debug=debug_multi, tag="inf_sign") test_signaling_nan = Test( vx, specifier=Test.IsSignalingNaN, debug=debug_multi, tag="is_signaling_nan") return_snan = Statement( ExpRaiseReturn(ML_FPE_Invalid, return_value=FP_QNaN(self.precision)) ) # return in case of infinity input infty_return = Statement( ConditionBlock( test_positive, Return(FP_PlusInfty(self.precision), precision=self.precision), Return(FP_PlusZero(self.precision), precision=self.precision) ) ) # return in case of specific value input (NaN or inf) specific_return = ConditionBlock( test_nan, ConditionBlock( test_signaling_nan, return_snan, Return(FP_QNaN(self.precision), precision=self.precision) ), infty_return) # return in case of standard (non-special) input # exclusion of early overflow and underflow cases precision_emax = self.precision.get_emax() precision_max_value = S2 * S2**precision_emax exp_overflow_bound = sollya.ceil(log(precision_max_value)) early_overflow_test = Comparison( vx, exp_overflow_bound, likely=False, specifier=Comparison.Greater) early_overflow_return = Statement( ClearException() if self.libm_compliant else Statement(), ExpRaiseReturn( ML_FPE_Inexact, ML_FPE_Overflow, return_value=FP_PlusInfty(self.precision) ) ) precision_emin = self.precision.get_emin_subnormal() precision_min_value = S2 ** precision_emin exp_underflow_bound = floor(log(precision_min_value)) early_underflow_test = Comparison( vx, exp_underflow_bound, likely=False, specifier=Comparison.Less) early_underflow_return = Statement( ClearException() if self.libm_compliant else Statement(), ExpRaiseReturn( ML_FPE_Inexact, ML_FPE_Underflow, return_value=FP_PlusZero(self.precision))) # constant computation invlog2 = self.precision.round_sollya_object(1/log(2), sollya.RN) interval_vx = Interval(exp_underflow_bound, exp_overflow_bound) interval_fk = interval_vx * invlog2 interval_k = Interval(floor(inf(interval_fk)), sollya.ceil(sup(interval_fk))) log2_hi_precision = self.precision.get_field_size() - (sollya.ceil(log2(sup(abs(interval_k)))) + 2) Log.report(Log.Info, "log2_hi_precision: %d" % log2_hi_precision) invlog2_cst = Constant(invlog2, precision = self.precision) log2_hi = round(log(2), log2_hi_precision, sollya.RN) log2_lo = self.precision.round_sollya_object(log(2) - log2_hi, sollya.RN) # argument reduction unround_k = vx * invlog2 unround_k.set_attributes(tag = "unround_k", debug = debug_multi) k = NearestInteger(unround_k, precision = self.precision, debug = debug_multi, tag="k") ik = NearestInteger(unround_k, precision = self.precision.get_integer_format(), debug = debug_multi, tag="ik") exact_pre_mul = (k * log2_hi) exact_pre_mul.set_attributes(exact= True) exact_hi_part = vx - exact_pre_mul exact_hi_part.set_attributes(exact = True, tag = "exact_hi", debug = debug_multi, prevent_optimization = True) exact_lo_part = - k * log2_lo exact_lo_part.set_attributes(tag = "exact_lo", debug = debug_multi, prevent_optimization = True) r = exact_hi_part + exact_lo_part r.set_tag("r") r.set_attributes(debug = debug_multi) approx_interval = Interval(-log(2)/2, log(2)/2) approx_interval_half = approx_interval / 2 approx_interval_split = [Interval(-log(2)/2, inf(approx_interval_half)), approx_interval_half, Interval(sup(approx_interval_half), log(2)/2)] # TODO: should be computed automatically exact_hi_interval = approx_interval exact_lo_interval = - interval_k * log2_lo opt_r = self.optimise_scheme(r, copy = {}) tag_map = {} self.opt_engine.register_nodes_by_tag(opt_r, tag_map) cg_eval_error_copy_map = { vx: Variable("x", precision=self.precision, interval=interval_vx), tag_map["k"]: Variable("k", interval=interval_k, precision=self.precision) } #try: if gappa_utils.is_gappa_installed(): eval_error = self.gappa_engine.get_eval_error_v2( self.opt_engine, opt_r, cg_eval_error_copy_map, gappa_filename=gappa_utils.generate_gappa_filename("red_arg.g")) else: eval_error = 0.0 Log.report(Log.Warning, "gappa is not installed in this environnement") Log.report(Log.Info, "eval error: %s" % eval_error) local_ulp = sup(ulp(sollya.exp(approx_interval), self.precision)) # FIXME refactor error_goal from accuracy Log.report(Log.Info, "accuracy: %s" % self.accuracy) if isinstance(self.accuracy, ML_Faithful): error_goal = local_ulp elif isinstance(self.accuracy, ML_CorrectlyRounded): error_goal = S2**-1 * local_ulp elif isinstance(self.accuracy, ML_DegradedAccuracyAbsolute): error_goal = self.accuracy.goal elif isinstance(self.accuracy, ML_DegradedAccuracyRelative): error_goal = self.accuracy.goal else: Log.report(Log.Error, "unknown accuracy: %s" % self.accuracy) # error_goal = local_ulp #S2**-(self.precision.get_field_size()+1) error_goal_approx = S2**-1 * error_goal Log.report(Log.Info, "\033[33;1m building mathematical polynomial \033[0m\n") poly_degree = max(sup(guessdegree(expm1(sollya.x)/sollya.x, approx_interval, error_goal_approx)) - 1, 2) init_poly_degree = poly_degree error_function = lambda p, f, ai, mod, t: dirtyinfnorm(f - p, ai) polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_estrin_scheme #polynomial_scheme_builder = PolynomialSchemeEvaluator.generate_horner_scheme MAX_NUM_ITERATION = 20 for _ in range(MAX_NUM_ITERATION): Log.report(Log.Info, "attempting poly degree: %d" % poly_degree) precision_list = [1] + [self.precision] * (poly_degree) poly_object, poly_approx_error = Polynomial.build_from_approximation_with_error(expm1(sollya.x), poly_degree, precision_list, approx_interval, sollya.absolute, error_function = error_function) Log.report(Log.Info, "polynomial: %s " % poly_object) sub_poly = poly_object.sub_poly(start_index = 2) Log.report(Log.Info, "polynomial: %s " % sub_poly) Log.report(Log.Info, "poly approx error: %s" % poly_approx_error) Log.report(Log.Info, "\033[33;1m generating polynomial evaluation scheme \033[0m") pre_poly = polynomial_scheme_builder(poly_object, r, unified_precision = self.precision) pre_poly.set_attributes(tag = "pre_poly", debug = debug_multi) pre_sub_poly = polynomial_scheme_builder(sub_poly, r, unified_precision = self.precision) pre_sub_poly.set_attributes(tag = "pre_sub_poly", debug = debug_multi) poly = 1 + (exact_hi_part + (exact_lo_part + pre_sub_poly)) poly.set_tag("poly") # optimizing poly before evaluation error computation #opt_poly = self.opt_engine.optimization_process(poly, self.precision, fuse_fma = fuse_fma) #opt_sub_poly = self.opt_engine.optimization_process(pre_sub_poly, self.precision, fuse_fma = fuse_fma) opt_poly = self.optimise_scheme(poly) opt_sub_poly = self.optimise_scheme(pre_sub_poly) # evaluating error of the polynomial approximation r_gappa_var = Variable("r", precision = self.precision, interval = approx_interval) exact_hi_gappa_var = Variable("exact_hi", precision = self.precision, interval = exact_hi_interval) exact_lo_gappa_var = Variable("exact_lo", precision = self.precision, interval = exact_lo_interval) vx_gappa_var = Variable("x", precision = self.precision, interval = interval_vx) k_gappa_var = Variable("k", interval = interval_k, precision = self.precision) #print "exact_hi interval: ", exact_hi_interval sub_poly_error_copy_map = { #r.get_handle().get_node(): r_gappa_var, #vx.get_handle().get_node(): vx_gappa_var, exact_hi_part.get_handle().get_node(): exact_hi_gappa_var, exact_lo_part.get_handle().get_node(): exact_lo_gappa_var, #k.get_handle().get_node(): k_gappa_var, } poly_error_copy_map = { exact_hi_part.get_handle().get_node(): exact_hi_gappa_var, exact_lo_part.get_handle().get_node(): exact_lo_gappa_var, } if gappa_utils.is_gappa_installed(): sub_poly_eval_error = -1.0 gappa_sub_poly_filename = gappa_utils.generate_gappa_filename("{}_gappa_sub_poly.g".format(self.function_name)) sub_poly_eval_error = self.gappa_engine.get_eval_error_v2(self.opt_engine, opt_sub_poly, sub_poly_error_copy_map, gappa_filename =gappa_sub_poly_filename) dichotomy_map = [ { exact_hi_part.get_handle().get_node(): approx_interval_split[0], }, { exact_hi_part.get_handle().get_node(): approx_interval_split[1], }, { exact_hi_part.get_handle().get_node(): approx_interval_split[2], }, ] gappa_poly_filename = gappa_utils.generate_gappa_filename("gappa_poly.g") poly_eval_error_dico = self.gappa_engine.get_eval_error_v3(self.opt_engine, opt_poly, poly_error_copy_map, gappa_filename=gappa_poly_filename, dichotomy = dichotomy_map) poly_eval_error = max([sup(abs(err)) for err in poly_eval_error_dico]) else: poly_eval_error = 0.0 sub_poly_eval_error = 0.0 Log.report(Log.Warning, "gappa is not installed in this environnement") Log.report(Log.Info, "stopping autonomous degree research") # incrementing polynomial degree to counteract initial decrementation effect poly_degree += 1 break Log.report(Log.Info, "poly evaluation error: %s" % poly_eval_error) Log.report(Log.Info, "sub poly evaluation error: %s" % sub_poly_eval_error) global_poly_error = None global_rel_poly_error = None for case_index in range(3): poly_error = poly_approx_error + poly_eval_error_dico[case_index] rel_poly_error = sup(abs(poly_error / sollya.exp(approx_interval_split[case_index]))) if global_rel_poly_error == None or rel_poly_error > global_rel_poly_error: global_rel_poly_error = rel_poly_error global_poly_error = poly_error flag = error_goal > global_rel_poly_error if flag: break else: poly_degree += 1 late_overflow_test = Comparison( ik, self.precision.get_emax(), specifier=Comparison.Greater, likely=False, debug=debug_multi, tag="late_overflow_test") overflow_exp_offset = int(self.precision.get_emax() - self.precision.get_field_size() / 2) cst_overflow_exp_offset = Constant(overflow_exp_offset, precision=self.precision.get_integer_format()) diff_k = Subtraction( ik, cst_overflow_exp_offset, precision=self.precision.get_integer_format(), debug=debug_multi, tag="diff_k", ) late_overflow_result = (ExponentInsertion(diff_k, precision = self.precision) * poly) * ExponentInsertion(cst_overflow_exp_offset, precision = self.precision) late_overflow_result.set_attributes(silent = False, tag = "late_overflow_result", debug = debug_multi, precision = self.precision) late_overflow_return = ConditionBlock(Test(late_overflow_result, specifier = Test.IsInfty, likely = False), ExpRaiseReturn(ML_FPE_Overflow, return_value = FP_PlusInfty(self.precision)), Return(late_overflow_result, precision=self.precision)) late_underflow_test = Comparison(k, self.precision.get_emin_normal(), specifier = Comparison.LessOrEqual, likely=False, tag="late_underflow_test") underflow_exp_offset = 2 * self.precision.get_field_size() corrected_exp = Addition( ik, Constant( underflow_exp_offset, precision=self.precision.get_integer_format() ), precision=self.precision.get_integer_format(), tag="corrected_exp" ) late_underflow_result = (ExponentInsertion(corrected_exp, precision = self.precision) * poly) * ExponentInsertion(-underflow_exp_offset, precision = self.precision) late_underflow_result.set_attributes(debug = debug_multi, tag = "late_underflow_result", silent = False) test_subnormal = Test(late_underflow_result, specifier = Test.IsSubnormal) late_underflow_return = Statement(ConditionBlock(test_subnormal, ExpRaiseReturn(ML_FPE_Underflow, return_value = late_underflow_result)), Return(late_underflow_result, precision=self.precision)) twok = ExponentInsertion(ik, tag = "exp_ik", debug = debug_multi, precision = self.precision) #std_result = twok * ((1 + exact_hi_part * pre_poly) + exact_lo_part * pre_poly) std_result = twok * poly std_result.set_attributes(tag = "std_result", debug = debug_multi) std_cond = LogicalNot(LogicalOr(late_overflow_test, late_underflow_test), likely=True) result_scheme = ConditionBlock( std_cond, Return(std_result, precision=self.precision), ConditionBlock( late_overflow_test, late_overflow_return, late_underflow_return, ) ) std_return = ConditionBlock(early_overflow_test, early_overflow_return, ConditionBlock(early_underflow_test, early_underflow_return, result_scheme)) # main scheme Log.report(Log.Info, "\033[33;1m MDL scheme \033[0m") scheme = ConditionBlock( test_nan_or_inf, Statement( ClearException() if self.libm_compliant else Statement(), specific_return ), std_return ) return scheme
def generate_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 generic_atan2_generate(self, _vx, vy=None): """ if vy is None, compute atan(_vx), else compute atan2(vy / vx) """ if vy is None: # approximation # if abs_vx <= 1.0 then atan(abx_vx) is directly approximated # if abs_vx > 1.0 then atan(abs_vx) = pi/2 - atan(1 / abs_vx) # # for vx >= 0, atan(vx) = atan(abs_vx) # # for vx < 0, atan(vx) = -atan(abs_vx) for vx < 0 # = -pi/2 + atan(1 / abs_vx) vx = _vx sign_cond = vx < 0 abs_vx = Select(vx < 0, -vx, vx, tag="abs_vx", debug=debug_multi) bound_cond = abs_vx > 1 inv_abs_vx = 1 / abs_vx # condition to select subtraction cond = LogicalOr(LogicalAnd(vx < 0, LogicalNot(bound_cond)), vx > 1, tag="cond", debug=debug_multi) # reduced argument red_vx = Select(bound_cond, inv_abs_vx, abs_vx, tag="red_vx", debug=debug_multi) offset = None else: # bound_cond is True iff Abs(vy / _vx) > 1.0 bound_cond = Abs(vy) > Abs(_vx) bound_cond.set_attributes(tag="bound_cond", debug=debug_multi) # vx and vy are of opposite signs #sign_cond = (_vx * vy) < 0 # using cast to int(signed) and bitwise xor # to determine if _vx and vy are of opposite sign rapidly fast_sign_cond = BitLogicXor( TypeCast(_vx, precision=self.precision.get_integer_format()), TypeCast(vy, precision=self.precision.get_integer_format()), precision=self.precision.get_integer_format()) < 0 # sign_cond = (_vx * vy) < 0 sign_cond = fast_sign_cond sign_cond.set_attributes(tag="sign_cond", debug=debug_multi) # condition to select subtraction # TODO: could be accelerated if LogicalXor existed slow_cond = LogicalOr( LogicalAnd(sign_cond, LogicalNot(bound_cond)), # 1 < (vy / _vx) < 0 LogicalAnd(bound_cond, LogicalNot(sign_cond)), # (vy / _vx) > 1 tag="cond", debug=debug_multi) cond = slow_cond numerator = Select(bound_cond, _vx, vy, tag="numerator", debug=debug_multi) denominator = Select(bound_cond, vy, _vx, tag="denominator", debug=debug_multi) # reduced argument red_vx = Abs(numerator) / Abs(denominator) red_vx.set_attributes(tag="red_vx", debug=debug_multi) offset = Select( _vx > 0, Constant(0, precision=self.precision), # vx < 0 Select( sign_cond, # vy > 0 Constant(sollya.pi, precision=self.precision), Constant(-sollya.pi, precision=self.precision), precision=self.precision), precision=self.precision, tag="offset") approx_fct = sollya.atan(sollya.x) if self.method == "piecewise": sign_vx = Select(cond, -1, 1, precision=self.precision, tag="sign_vx", debug=debug_multi) cst_sign = Select(sign_cond, -1, 1, precision=self.precision, tag="cst_sign", debug=debug_multi) cst = cst_sign * Select( bound_cond, sollya.pi / 2, 0, precision=self.precision) cst.set_attributes(tag="cst", debug=debug_multi) bound_low = 0.0 bound_high = 1.0 num_intervals = self.num_sub_intervals error_threshold = S2**-(self.precision.get_mantissa_size() + 8) approx, eval_error = piecewise_approximation( approx_fct, red_vx, self.precision, bound_low=bound_low, bound_high=bound_high, max_degree=None, num_intervals=num_intervals, error_threshold=error_threshold, odd=True) result = cst + sign_vx * approx result.set_attributes(tag="result", precision=self.precision, debug=debug_multi) elif self.method == "single": approx_interval = Interval(0, 1.0) # determining the degree of the polynomial approximation poly_degree_range = sollya.guessdegree( approx_fct / sollya.x, approx_interval, S2**-(self.precision.get_field_size() + 2)) poly_degree = int(sollya.sup(poly_degree_range)) + 4 Log.report(Log.Info, "poly_degree={}".format(poly_degree)) # arctan is an odd function, so only odd coefficient must be non-zero poly_degree_list = list(range(1, poly_degree + 1, 2)) poly_object, poly_error = Polynomial.build_from_approximation_with_error( approx_fct, poly_degree_list, [1] + [self.precision.get_sollya_object()] * (len(poly_degree_list) - 1), approx_interval) odd_predicate = lambda index, _: ((index - 1) % 4 != 0) even_predicate = lambda index, _: (index != 1 and (index - 1) % 4 == 0) poly_odd_object = poly_object.sub_poly_cond(odd_predicate, offset=1) poly_even_object = poly_object.sub_poly_cond(even_predicate, offset=1) sollya.settings.display = sollya.hexadecimal Log.report(Log.Info, "poly_error: {}".format(poly_error)) Log.report(Log.Info, "poly_odd: {}".format(poly_odd_object)) Log.report(Log.Info, "poly_even: {}".format(poly_even_object)) poly_odd = PolynomialSchemeEvaluator.generate_horner_scheme( poly_odd_object, abs_vx) poly_odd.set_attributes(tag="poly_odd", debug=debug_multi) poly_even = PolynomialSchemeEvaluator.generate_horner_scheme( poly_even_object, abs_vx) poly_even.set_attributes(tag="poly_even", debug=debug_multi) exact_sum = poly_odd + poly_even exact_sum.set_attributes(tag="exact_sum", debug=debug_multi) # poly_even should be (1 + poly_even) result = vx + vx * exact_sum result.set_attributes(tag="result", precision=self.precision, debug=debug_multi) else: raise NotImplementedError if not offset is None: result = result + offset std_scheme = Statement(Return(result)) scheme = std_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): # declaring target and instantiating optimization engine vx = self.implementation.add_input_variable("x", self.precision) vx.set_attributes(precision=self.precision, tag="vx", debug=debug_multi) Log.set_dump_stdout(True) Log.report(Log.Info, "\033[33;1m Generating implementation scheme \033[0m") if self.debug_flag: Log.report(Log.Info, "\033[31;1m debug has been enabled \033[0;m") C0 = Constant(0, precision=self.precision) C0_plus = Constant(FP_PlusZero(self.precision)) C0_minus = Constant(FP_MinusZero(self.precision)) def local_test(specifier, tag): """ Local wrapper to generate Test operations """ return Test(vx, specifier=specifier, likely=False, debug=debug_multi, tag="is_%s" % tag, precision=ML_Bool) test_NaN = local_test(Test.IsNaN, "is_NaN") test_inf = local_test(Test.IsInfty, "is_Inf") test_NaN_or_Inf = local_test(Test.IsInfOrNaN, "is_Inf_Or_Nan") test_negative = Comparison(vx, C0, specifier=Comparison.Less, debug=debug_multi, tag="is_Negative", precision=ML_Bool, likely=False) test_NaN_or_Neg = LogicalOr(test_NaN, test_negative, precision=ML_Bool) test_std = LogicalNot(LogicalOr(test_NaN_or_Inf, test_negative, precision=ML_Bool, likely=False), precision=ML_Bool, likely=True) test_zero = Comparison(vx, C0, specifier=Comparison.Equal, likely=False, debug=debug_multi, tag="Is_Zero", precision=ML_Bool) return_NaN_or_neg = Statement(Return(FP_QNaN(self.precision))) return_inf = Statement(Return(FP_PlusInfty(self.precision))) return_PosZero = Return(C0_plus) return_NegZero = Return(C0_minus) NR_init = ReciprocalSquareRootSeed(vx, precision=self.precision, tag="sqrt_seed", debug=debug_multi) result = compute_sqrt(vx, NR_init, int(self.num_iter), precision=self.precision) return_non_std = ConditionBlock( test_NaN_or_Neg, return_NaN_or_neg, ConditionBlock( test_inf, return_inf, ConditionBlock(test_zero, return_PosZero, return_NegZero))) return_std = Return(result) scheme = ConditionBlock(test_std, return_std, return_non_std) return scheme
def booth_radix4_multiply(lhs, rhs, pos_bit_heap, neg_bit_heap): """ Compute the multiplication @p lhs x @p rhs using radix 4 Booth recoding and drop the generated partial product in @p pos_bit_heap and @p neg_bit_heap based on their sign """ # booth recoded partial product for n-th digit # is based on digit from n-1 to n+1 # (n+1) | (n) | (n-1) | PP | # ------|-----|-------|------| # 0 | 0 | 0 | +0 | # 0 | 0 | 1 | +X | # 0 | 1 | 0 | +X | # 0 | 1 | 1 | +2x | # 1 | 0 | 0 | -2X | # 1 | 0 | 1 | -X | # 1 | 1 | 0 | -X | # 1 | 1 | 1 | +0 | # ------|-----|-------|------| assert lhs.get_precision().get_bit_size() >= 2 # lhs is the recoded operand # RECODING DIGITS # first recoded digit is padded right by 0 first_digit = Concatenation(SubSignalSelection( lhs, 0, 1, precision=ML_StdLogicVectorFormat(2)), Constant(0, precision=ML_StdLogic), precision=ML_StdLogicVectorFormat(3), debug=debug_std, tag="booth_digit_0") digit_list = [(first_digit, 0)] for digit_index in range(2, lhs.get_precision().get_bit_size(), 2): if digit_index + 1 < lhs.get_precision().get_bit_size(): # digits exist completely in lhs digit = SubSignalSelection(lhs, digit_index - 1, digit_index + 1, tag="booth_digit_%d" % digit_index, debug=debug_std) else: # MSB padding required sign_ext = Constant(0, precision=ML_StdLogic) if not ( lhs.get_precision().get_signed()) else BitSelection( lhs, lhs.get_precision().get_bit_size() - 1) digit = Concatenation(sign_ext, SubSignalSelection(lhs, digit_index - 1, digit_index), precision=ML_StdLogicVectorFormat(3), debug=debug_std, tag="booth_digit_%d" % digit_index) digit_list.append((digit, digit_index)) # if lhs size is a mutiple of two and it is unsigned # than an extra digit must be generated to ensure a positive result if lhs.get_precision().get_bit_size() % 2 == 0 and not ( lhs.get_precision().get_signed()): digit_index = lhs.get_precision().get_bit_size() - 1 digit = Concatenation(Constant(0, precision=ML_StdLogicVectorFormat(2)), BitSelection(lhs, digit_index), precision=ML_StdLogicVectorFormat(3), debug=debug_std, tag="booth_digit_%d" % (digit_index + 1)) digit_list.append((digit, digit_index + 1)) def DCV(value): """ Digit Constante Value """ return Constant(value, precision=ML_StdLogicVectorFormat(3)) # PARTIAL PRODUCT GENERATION # Radix-4 booth recoding requires the following Partial Products # -2.rhs, -rhs, 0, rhs and 2.rhs # Negative PP are obtained by 1's complement of the value correctly shifted # adding a positive one to the LSB (inserted separately) and assuming # MSB digit has a negative weight for digit, index in digit_list: pp_zero = LogicalOr(Equal(digit, DCV(0), precision=ML_Bool), Equal(digit, DCV(7), precision=ML_Bool), precision=ML_Bool) pp_shifted = LogicalOr(Equal(digit, DCV(3), precision=ML_Bool), Equal(digit, DCV(4), precision=ML_Bool), precision=ML_Bool) # excluding zero case pp_neg_bit = BitSelection(digit, 2) pp_neg = equal_to(pp_neg_bit, 1) pp_neg_lsb_carryin = Select(LogicalAnd(pp_neg, LogicalNot(pp_zero)), Constant(1, precision=ML_StdLogic), Constant(0, precision=ML_StdLogic), tag="pp_%d_neg_lsb_carryin" % index, debug=debug_std) # LSB digit lsb_pp_digit = Select(pp_shifted, Constant(0, precision=ML_StdLogic), BitSelection(rhs, 0), precision=ML_StdLogic) lsb_local_pp = Select(pp_zero, Constant(0, precision=ML_StdLogic), Select(pp_neg, BitLogicNegate(lsb_pp_digit), lsb_pp_digit, precision=ML_StdLogic), debug=debug_std, tag="lsb_local_pp_%d" % index, precision=ML_StdLogic) pos_bit_heap.insert_bit(index, lsb_local_pp) pos_bit_heap.insert_bit(index, pp_neg_lsb_carryin) # other digits rhs_size = rhs.get_precision().get_bit_size() for k in range(1, rhs_size): pp_digit = Select(pp_shifted, BitSelection(rhs, k - 1), BitSelection(rhs, k), precision=ML_StdLogic) local_pp = Select(pp_zero, Constant(0, precision=ML_StdLogic), Select(pp_neg, BitLogicNegate(pp_digit), pp_digit, precision=ML_StdLogic), debug=debug_std, tag="local_pp_%d_%d" % (index, k), precision=ML_StdLogic) pos_bit_heap.insert_bit(index + k, local_pp) # MSB digit msb_pp_digit = pp_digit = Select( pp_shifted, BitSelection(rhs, rhs_size - 1), # TODO: fix for signed rhs Constant(0, precision=ML_StdLogic) if not (rhs.get_precision().get_signed()) else BitSelection( rhs, rhs_size - 1), precision=ML_StdLogic) msb_pp = Select(pp_zero, Constant(0, precision=ML_StdLogic), Select(pp_neg, BitLogicNegate(msb_pp_digit), msb_pp_digit, precision=ML_StdLogic), debug=debug_std, tag="msb_pp_%d" % (index), precision=ML_StdLogic) if rhs.get_precision().get_signed(): neg_bit_heap.insert_bit(index + rhs_size, msb_pp) else: pos_bit_heap.insert_bit(index + rhs_size, msb_pp) # MSB negative digit, # 'rhs_size + index) is the position of the MSB digit of rhs shifted by 1 # we add +1 to get to the sign position neg_bit_heap.insert_bit(index + rhs_size + 1, pp_neg_lsb_carryin)
def LogicalXor(a, b): return LogicalOr(LogicalAnd(a, LogicalNot(b)), LogicalAnd(LogicalNot(a), b))
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