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1/*2 * Double-precision log(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 T __log_data.tab15#define T2 __log_data.tab216#define B __log_data.poly117#define A __log_data.poly18#define Ln2hi __log_data.ln2hi19#define Ln2lo __log_data.ln2lo20#define N (1 << LOG_TABLE_BITS)21#define OFF 0x3fe600000000000022 23/* Top 16 bits of a double. */24static inline uint32_t25top16 (double x)26{27 return asuint64 (x) >> 48;28}29 30double31log (double x)32{33 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */34 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;35 uint64_t ix, iz, tmp;36 uint32_t top;37 int k, i;38 39 ix = asuint64 (x);40 top = top16 (x);41 42#if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 1143# define LO asuint64 (1.0 - 0x1p-5)44# define HI asuint64 (1.0 + 0x1.1p-5)45#elif LOG_POLY1_ORDER == 1246# define LO asuint64 (1.0 - 0x1p-4)47# define HI asuint64 (1.0 + 0x1.09p-4)48#endif49 if (unlikely (ix - LO < HI - LO))50 {51 /* Handle close to 1.0 inputs separately. */52 /* Fix sign of zero with downward rounding when x==1. */53 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))54 return 0;55 r = x - 1.0;56 r2 = r * r;57 r3 = r * r2;58#if LOG_POLY1_ORDER == 1059 /* Worst-case error is around 0.516 ULP. */60 y = r3 * (B[1] + r * B[2] + r2 * B[3]61 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));62 w = B[0] * r2; /* B[0] == -0.5. */63 hi = r + w;64 y += r - hi + w;65 y += hi;66#elif LOG_POLY1_ORDER == 1167 /* Worst-case error is around 0.516 ULP. */68 y = r3 * (B[1] + r * B[2]69 + r2 * (B[3] + r * B[4] + r2 * B[5]70 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));71 w = B[0] * r2; /* B[0] == -0.5. */72 hi = r + w;73 y += r - hi + w;74 y += hi;75#elif LOG_POLY1_ORDER == 1276 y = r3 * (B[1] + r * B[2] + r2 * B[3]77 + r3 * (B[4] + r * B[5] + r2 * B[6]78 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));79# if N <= 6480 /* Worst-case error is around 0.532 ULP. */81 w = B[0] * r2; /* B[0] == -0.5. */82 hi = r + w;83 y += r - hi + w;84 y += hi;85# else86 /* Worst-case error is around 0.507 ULP. */87 w = r * 0x1p27;88 double_t rhi = r + w - w;89 double_t rlo = r - rhi;90 w = rhi * rhi * B[0]; /* B[0] == -0.5. */91 hi = r + w;92 lo = r - hi + w;93 lo += B[0] * rlo * (rhi + r);94 y += lo;95 y += hi;96# endif97#endif98 return eval_as_double (y);99 }100 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))101 {102 /* x < 0x1p-1022 or inf or nan. */103 if (ix * 2 == 0)104 return __math_divzero (1);105 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */106 return x;107 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)108 return __math_invalid (x);109 /* x is subnormal, normalize it. */110 ix = asuint64 (x * 0x1p52);111 ix -= 52ULL << 52;112 }113 114 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.115 The range is split into N subintervals.116 The ith subinterval contains z and c is near its center. */117 tmp = ix - OFF;118 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;119 k = (int64_t) tmp >> 52; /* arithmetic shift */120 iz = ix - (tmp & 0xfffULL << 52);121 invc = T[i].invc;122 logc = T[i].logc;123 z = asdouble (iz);124 125 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */126 /* r ~= z/c - 1, |r| < 1/(2*N). */127#if HAVE_FAST_FMA128 /* rounding error: 0x1p-55/N. */129 r = fma (z, invc, -1.0);130#else131 /* rounding error: 0x1p-55/N + 0x1p-66. */132 r = (z - T2[i].chi - T2[i].clo) * invc;133#endif134 kd = (double_t) k;135 136 /* hi + lo = r + log(c) + k*Ln2. */137 w = kd * Ln2hi + logc;138 hi = w + r;139 lo = w - hi + r + kd * Ln2lo;140 141 /* log(x) = lo + (log1p(r) - r) + hi. */142 r2 = r * r; /* rounding error: 0x1p-54/N^2. */143 /* Worst case error if |y| > 0x1p-5:144 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)145 Worst case error if |y| > 0x1p-4:146 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */147#if LOG_POLY_ORDER == 6148 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;149#elif LOG_POLY_ORDER == 7150 y = lo151 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])152 + r2 * r2 * (A[4] + r * A[5]))153 + hi;154#endif155 return eval_as_double (y);156}157#if USE_GLIBC_ABI158strong_alias (log, __log_finite)159hidden_alias (log, __ieee754_log)160# if LDBL_MANT_DIG == 53161long double logl (long double x) { return log (x); }162# endif163#endif164