1/* 2 * Double-precision log(x) function. 3 * 4 * Copyright (c) 2018-2019, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8#include <float.h> 9#include <math.h> 10#include <stdint.h> 11#include "math_config.h" 12 13#define T __log_data.tab 14#define T2 __log_data.tab2 15#define B __log_data.poly1 16#define A __log_data.poly 17#define Ln2hi __log_data.ln2hi 18#define Ln2lo __log_data.ln2lo 19#define N (1 << LOG_TABLE_BITS) 20#define OFF 0x3fe6000000000000 21 22/* Top 16 bits of a double. */ 23static inline uint32_t 24top16 (double x) 25{ 26 return asuint64 (x) >> 48; 27} 28 29double 30log (double x) 31{ 32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 33 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; 34 uint64_t ix, iz, tmp; 35 uint32_t top; 36 int k, i; 37 38 ix = asuint64 (x); 39 top = top16 (x); 40 41#if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11 42# define LO asuint64 (1.0 - 0x1p-5) 43# define HI asuint64 (1.0 + 0x1.1p-5) 44#elif LOG_POLY1_ORDER == 12 45# define LO asuint64 (1.0 - 0x1p-4) 46# define HI asuint64 (1.0 + 0x1.09p-4) 47#endif 48 if (unlikely (ix - LO < HI - LO)) 49 { 50 /* Handle close to 1.0 inputs separately. */ 51 /* Fix sign of zero with downward rounding when x==1. */ 52 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 53 return 0; 54 r = x - 1.0; 55 r2 = r * r; 56 r3 = r * r2; 57#if LOG_POLY1_ORDER == 10 58 /* Worst-case error is around 0.516 ULP. */ 59 y = r3 * (B[1] + r * B[2] + r2 * B[3] 60 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8]))); 61 w = B[0] * r2; /* B[0] == -0.5. */ 62 hi = r + w; 63 y += r - hi + w; 64 y += hi; 65#elif LOG_POLY1_ORDER == 11 66 /* Worst-case error is around 0.516 ULP. */ 67 y = r3 * (B[1] + r * B[2] 68 + r2 * (B[3] + r * B[4] + r2 * B[5] 69 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9]))); 70 w = B[0] * r2; /* B[0] == -0.5. */ 71 hi = r + w; 72 y += r - hi + w; 73 y += hi; 74#elif LOG_POLY1_ORDER == 12 75 y = r3 * (B[1] + r * B[2] + r2 * B[3] 76 + r3 * (B[4] + r * B[5] + r2 * B[6] 77 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); 78# if N <= 64 79 /* Worst-case error is around 0.532 ULP. */ 80 w = B[0] * r2; /* B[0] == -0.5. */ 81 hi = r + w; 82 y += r - hi + w; 83 y += hi; 84# else 85 /* Worst-case error is around 0.507 ULP. */ 86 w = r * 0x1p27; 87 double_t rhi = r + w - w; 88 double_t rlo = r - rhi; 89 w = rhi * rhi * B[0]; /* B[0] == -0.5. */ 90 hi = r + w; 91 lo = r - hi + w; 92 lo += B[0] * rlo * (rhi + r); 93 y += lo; 94 y += hi; 95# endif 96#endif 97 return eval_as_double (y); 98 } 99 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 100 { 101 /* x < 0x1p-1022 or inf or nan. */ 102 if (ix * 2 == 0) 103 return __math_divzero (1); 104 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 105 return x; 106 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 107 return __math_invalid (x); 108 /* x is subnormal, normalize it. */ 109 ix = asuint64 (x * 0x1p52); 110 ix -= 52ULL << 52; 111 } 112 113 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 114 The range is split into N subintervals. 115 The ith subinterval contains z and c is near its center. */ 116 tmp = ix - OFF; 117 i = (tmp >> (52 - LOG_TABLE_BITS)) % N; 118 k = (int64_t) tmp >> 52; /* arithmetic shift */ 119 iz = ix - (tmp & 0xfffULL << 52); 120 invc = T[i].invc; 121 logc = T[i].logc; 122 z = asdouble (iz); 123 124 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ 125 /* r ~= z/c - 1, |r| < 1/(2*N). */ 126#if HAVE_FAST_FMA 127 /* rounding error: 0x1p-55/N. */ 128 r = fma (z, invc, -1.0); 129#else 130 /* rounding error: 0x1p-55/N + 0x1p-66. */ 131 r = (z - T2[i].chi - T2[i].clo) * invc; 132#endif 133 kd = (double_t) k; 134 135 /* hi + lo = r + log(c) + k*Ln2. */ 136 w = kd * Ln2hi + logc; 137 hi = w + r; 138 lo = w - hi + r + kd * Ln2lo; 139 140 /* log(x) = lo + (log1p(r) - r) + hi. */ 141 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 142 /* Worst case error if |y| > 0x1p-5: 143 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma) 144 Worst case error if |y| > 0x1p-4: 145 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */ 146#if LOG_POLY_ORDER == 6 147 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; 148#elif LOG_POLY_ORDER == 7 149 y = lo 150 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3]) 151 + r2 * r2 * (A[4] + r * A[5])) 152 + hi; 153#endif 154 return eval_as_double (y); 155} 156#if USE_GLIBC_ABI 157strong_alias (log, __log_finite) 158hidden_alias (log, __ieee754_log) 159# if LDBL_MANT_DIG == 53 160long double logl (long double x) { return log (x); } 161# endif 162#endif 163