1/* Double-precision log(x) function.
2 Copyright (C) 2018-2020 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <stdint.h>
21#include <math-svid-compat.h>
22#include <libm-alias-finite.h>
23#include <libm-alias-double.h>
24#include "math_config.h"
25
26#define T __log_data.tab
27#define T2 __log_data.tab2
28#define B __log_data.poly1
29#define A __log_data.poly
30#define Ln2hi __log_data.ln2hi
31#define Ln2lo __log_data.ln2lo
32#define N (1 << LOG_TABLE_BITS)
33#define OFF 0x3fe6000000000000
34
35/* Top 16 bits of a double. */
36static inline uint32_t
37top16 (double x)
38{
39 return asuint64 (x) >> 48;
40}
41
42#ifndef SECTION
43# define SECTION
44#endif
45
46double
47SECTION
48__log (double x)
49{
50 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
51 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
52 uint64_t ix, iz, tmp;
53 uint32_t top;
54 int k, i;
55
56 ix = asuint64 (x);
57 top = top16 (x);
58
59#define LO asuint64 (1.0 - 0x1p-4)
60#define HI asuint64 (1.0 + 0x1.09p-4)
61 if (__glibc_unlikely (ix - LO < HI - LO))
62 {
63 /* Handle close to 1.0 inputs separately. */
64 /* Fix sign of zero with downward rounding when x==1. */
65 if (WANT_ROUNDING && __glibc_unlikely (ix == asuint64 (1.0)))
66 return 0;
67 r = x - 1.0;
68 r2 = r * r;
69 r3 = r * r2;
70 y = r3 * (B[1] + r * B[2] + r2 * B[3]
71 + r3 * (B[4] + r * B[5] + r2 * B[6]
72 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
73 /* Worst-case error is around 0.507 ULP. */
74 w = r * 0x1p27;
75 double_t rhi = r + w - w;
76 double_t rlo = r - rhi;
77 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
78 hi = r + w;
79 lo = r - hi + w;
80 lo += B[0] * rlo * (rhi + r);
81 y += lo;
82 y += hi;
83 return y;
84 }
85 if (__glibc_unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
86 {
87 /* x < 0x1p-1022 or inf or nan. */
88 if (ix * 2 == 0)
89 return __math_divzero (1);
90 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
91 return x;
92 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
93 return __math_invalid (x);
94 /* x is subnormal, normalize it. */
95 ix = asuint64 (x * 0x1p52);
96 ix -= 52ULL << 52;
97 }
98
99 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
100 The range is split into N subintervals.
101 The ith subinterval contains z and c is near its center. */
102 tmp = ix - OFF;
103 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
104 k = (int64_t) tmp >> 52; /* arithmetic shift */
105 iz = ix - (tmp & 0xfffULL << 52);
106 invc = T[i].invc;
107 logc = T[i].logc;
108 z = asdouble (iz);
109
110 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
111 /* r ~= z/c - 1, |r| < 1/(2*N). */
112#ifdef __FP_FAST_FMA
113 /* rounding error: 0x1p-55/N. */
114 r = __builtin_fma (z, invc, -1.0);
115#else
116 /* rounding error: 0x1p-55/N + 0x1p-66. */
117 r = (z - T2[i].chi - T2[i].clo) * invc;
118#endif
119 kd = (double_t) k;
120
121 /* hi + lo = r + log(c) + k*Ln2. */
122 w = kd * Ln2hi + logc;
123 hi = w + r;
124 lo = w - hi + r + kd * Ln2lo;
125
126 /* log(x) = lo + (log1p(r) - r) + hi. */
127 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
128 /* Worst case error if |y| > 0x1p-4: 0.519 ULP (0.520 ULP without fma).
129 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
130 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
131 return y;
132}
133#ifndef __log
134strong_alias (__log, __ieee754_log)
135libm_alias_finite (__ieee754_log, __log)
136# if LIBM_SVID_COMPAT
137versioned_symbol (libm, __log, log, GLIBC_2_29);
138libm_alias_double_other (__log, log)
139# else
140libm_alias_double (__log, log)
141# endif
142#endif
143