1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177
|
/*
* Double-precision e^x function.
*
* Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
* See https://llvm.org/LICENSE.txt for license information.
* SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
*/
#include <float.h>
#include <math.h>
#include <stdint.h>
#include "math_config.h"
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
/* Handle cases that may overflow or underflow when computing the result that
is scale*(1+TMP) without intermediate rounding. The bit representation of
scale is in SBITS, however it has a computed exponent that may have
overflown into the sign bit so that needs to be adjusted before using it as
a double. (int32_t)KI is the k used in the argument reduction and exponent
adjustment of scale, positive k here means the result may overflow and
negative k means the result may underflow. */
static inline double
specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0)
{
/* k > 0, the exponent of scale might have overflowed by <= 460. */
sbits -= 1009ull << 52;
scale = asdouble (sbits);
y = 0x1p1009 * (scale + scale * tmp);
return check_oflow (eval_as_double (y));
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
scale = asdouble (sbits);
y = scale + scale * tmp;
if (y < 1.0)
{
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo;
lo = scale - y + scale * tmp;
hi = 1.0 + y;
lo = 1.0 - hi + y + lo;
y = eval_as_double (hi + lo) - 1.0;
/* Avoid -0.0 with downward rounding. */
if (WANT_ROUNDING && y == 0.0)
y = 0.0;
/* The underflow exception needs to be signaled explicitly. */
force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return check_uflow (eval_as_double (y));
}
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t
top12 (double x)
{
return asuint64 (x) >> 52;
}
/* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
If hastail is 0 then xtail is assumed to be 0 too. */
static inline double
exp_inline (double x, double xtail, int hastail)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, scale, tail, tmp;
abstop = top12 (x) & 0x7ff;
if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
{
if (abstop - top12 (0x1p-54) >= 0x80000000)
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
return WANT_ROUNDING ? 1.0 + x : 1.0;
if (abstop >= top12 (1024.0))
{
if (asuint64 (x) == asuint64 (-INFINITY))
return 0.0;
if (abstop >= top12 (INFINITY))
return 1.0 + x;
if (asuint64 (x) >> 63)
return __math_uflow (0);
else
return __math_oflow (0);
}
/* Large x is special cased below. */
abstop = 0;
}
/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
z = InvLn2N * x;
#if TOINT_INTRINSICS
kd = roundtoint (z);
ki = converttoint (z);
#elif EXP_USE_TOINT_NARROW
/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
kd = eval_as_double (z + Shift);
ki = asuint64 (kd) >> 16;
kd = (double_t) (int32_t) ki;
#else
/* z - kd is in [-1, 1] in non-nearest rounding modes. */
kd = eval_as_double (z + Shift);
ki = asuint64 (kd);
kd -= Shift;
#endif
r = x + kd * NegLn2hiN + kd * NegLn2loN;
/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
if (hastail)
r += xtail;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = ki << (52 - EXP_TABLE_BITS);
tail = asdouble (T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.25/N ulp larger. */
/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
#if EXP_POLY_ORDER == 4
tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
#elif EXP_POLY_ORDER == 5
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
#elif EXP_POLY_ORDER == 6
tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
#endif
if (unlikely (abstop == 0))
return specialcase (tmp, sbits, ki);
scale = asdouble (sbits);
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
is no spurious underflow here even without fma. */
return eval_as_double (scale + scale * tmp);
}
double
exp (double x)
{
return exp_inline (x, 0, 0);
}
/* May be useful for implementing pow where more than double
precision input is needed. */
double
__exp_dd (double x, double xtail)
{
return exp_inline (x, xtail, 1);
}
#if USE_GLIBC_ABI
strong_alias (exp, __exp_finite)
hidden_alias (exp, __ieee754_exp)
hidden_alias (__exp_dd, __exp1)
# if LDBL_MANT_DIG == 53
long double expl (long double x) { return exp (x); }
# endif
#endif
|