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1/*2 * Double-precision e^x function.3 *4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.5 * See https://llvm.org/LICENSE.txt for license information.6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception7 */8 9#include <float.h>10#include <math.h>11#include <stdint.h>12#include "math_config.h"13 14#define N (1 << EXP_TABLE_BITS)15#define InvLn2N __exp_data.invln2N16#define NegLn2hiN __exp_data.negln2hiN17#define NegLn2loN __exp_data.negln2loN18#define Shift __exp_data.shift19#define T __exp_data.tab20#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]21#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]22#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]23#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]24#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]25 26/* Handle cases that may overflow or underflow when computing the result that27 is scale*(1+TMP) without intermediate rounding. The bit representation of28 scale is in SBITS, however it has a computed exponent that may have29 overflown into the sign bit so that needs to be adjusted before using it as30 a double. (int32_t)KI is the k used in the argument reduction and exponent31 adjustment of scale, positive k here means the result may overflow and32 negative k means the result may underflow. */33static inline double34specialcase (double_t tmp, uint64_t sbits, uint64_t ki)35{36 double_t scale, y;37 38 if ((ki & 0x80000000) == 0)39 {40 /* k > 0, the exponent of scale might have overflowed by <= 460. */41 sbits -= 1009ull << 52;42 scale = asdouble (sbits);43 y = 0x1p1009 * (scale + scale * tmp);44 return check_oflow (eval_as_double (y));45 }46 /* k < 0, need special care in the subnormal range. */47 sbits += 1022ull << 52;48 scale = asdouble (sbits);49 y = scale + scale * tmp;50 if (y < 1.0)51 {52 /* Round y to the right precision before scaling it into the subnormal53 range to avoid double rounding that can cause 0.5+E/2 ulp error where54 E is the worst-case ulp error outside the subnormal range. So this55 is only useful if the goal is better than 1 ulp worst-case error. */56 double_t hi, lo;57 lo = scale - y + scale * tmp;58 hi = 1.0 + y;59 lo = 1.0 - hi + y + lo;60 y = eval_as_double (hi + lo) - 1.0;61 /* Avoid -0.0 with downward rounding. */62 if (WANT_ROUNDING && y == 0.0)63 y = 0.0;64 /* The underflow exception needs to be signaled explicitly. */65 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);66 }67 y = 0x1p-1022 * y;68 return check_uflow (eval_as_double (y));69}70 71/* Top 12 bits of a double (sign and exponent bits). */72static inline uint32_t73top12 (double x)74{75 return asuint64 (x) >> 52;76}77 78/* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.79 If hastail is 0 then xtail is assumed to be 0 too. */80static inline double81exp_inline (double x, double xtail, int hastail)82{83 uint32_t abstop;84 uint64_t ki, idx, top, sbits;85 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */86 double_t kd, z, r, r2, scale, tail, tmp;87 88 abstop = top12 (x) & 0x7ff;89 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))90 {91 if (abstop - top12 (0x1p-54) >= 0x80000000)92 /* Avoid spurious underflow for tiny x. */93 /* Note: 0 is common input. */94 return WANT_ROUNDING ? 1.0 + x : 1.0;95 if (abstop >= top12 (1024.0))96 {97 if (asuint64 (x) == asuint64 (-INFINITY))98 return 0.0;99 if (abstop >= top12 (INFINITY))100 return 1.0 + x;101 if (asuint64 (x) >> 63)102 return __math_uflow (0);103 else104 return __math_oflow (0);105 }106 /* Large x is special cased below. */107 abstop = 0;108 }109 110 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */111 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */112 z = InvLn2N * x;113#if TOINT_INTRINSICS114 kd = roundtoint (z);115 ki = converttoint (z);116#elif EXP_USE_TOINT_NARROW117 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */118 kd = eval_as_double (z + Shift);119 ki = asuint64 (kd) >> 16;120 kd = (double_t) (int32_t) ki;121#else122 /* z - kd is in [-1, 1] in non-nearest rounding modes. */123 kd = eval_as_double (z + Shift);124 ki = asuint64 (kd);125 kd -= Shift;126#endif127 r = x + kd * NegLn2hiN + kd * NegLn2loN;128 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */129 if (hastail)130 r += xtail;131 /* 2^(k/N) ~= scale * (1 + tail). */132 idx = 2 * (ki % N);133 top = ki << (52 - EXP_TABLE_BITS);134 tail = asdouble (T[idx]);135 /* This is only a valid scale when -1023*N < k < 1024*N. */136 sbits = T[idx + 1] + top;137 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */138 /* Evaluation is optimized assuming superscalar pipelined execution. */139 r2 = r * r;140 /* Without fma the worst case error is 0.25/N ulp larger. */141 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */142#if EXP_POLY_ORDER == 4143 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);144#elif EXP_POLY_ORDER == 5145 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);146#elif EXP_POLY_ORDER == 6147 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);148#endif149 if (unlikely (abstop == 0))150 return specialcase (tmp, sbits, ki);151 scale = asdouble (sbits);152 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there153 is no spurious underflow here even without fma. */154 return eval_as_double (scale + scale * tmp);155}156 157double158exp (double x)159{160 return exp_inline (x, 0, 0);161}162 163/* May be useful for implementing pow where more than double164 precision input is needed. */165double166__exp_dd (double x, double xtail)167{168 return exp_inline (x, xtail, 1);169}170#if USE_GLIBC_ABI171strong_alias (exp, __exp_finite)172hidden_alias (exp, __ieee754_exp)173hidden_alias (__exp_dd, __exp1)174# if LDBL_MANT_DIG == 53175long double expl (long double x) { return exp (x); }176# endif177#endif178