1 /* Double-precision log2(x) function.
2 Copyright (C) 2018-2021 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 __log2_data.tab
27 #define T2 __log2_data.tab2
28 #define B __log2_data.poly1
29 #define A __log2_data.poly
30 #define InvLn2hi __log2_data.invln2hi
31 #define InvLn2lo __log2_data.invln2lo
32 #define N (1 << LOG2_TABLE_BITS)
33 #define OFF 0x3fe6000000000000
34
35 /* Top 16 bits of a double. */
36 static inline uint32_t
top16(double x)37 top16 (double x)
38 {
39 return asuint64 (x) >> 48;
40 }
41
42 double
__log2(double x)43 __log2 (double x)
44 {
45 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
46 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
47 uint64_t ix, iz, tmp;
48 uint32_t top;
49 int k, i;
50
51 ix = asuint64 (x);
52 top = top16 (x);
53
54 #define LO asuint64 (1.0 - 0x1.5b51p-5)
55 #define HI asuint64 (1.0 + 0x1.6ab2p-5)
56 if (__glibc_unlikely (ix - LO < HI - LO))
57 {
58 /* Handle close to 1.0 inputs separately. */
59 /* Fix sign of zero with downward rounding when x==1. */
60 if (WANT_ROUNDING && __glibc_unlikely (ix == asuint64 (1.0)))
61 return 0;
62 r = x - 1.0;
63 #ifdef __FP_FAST_FMA
64 hi = r * InvLn2hi;
65 lo = r * InvLn2lo + __builtin_fma (r, InvLn2hi, -hi);
66 #else
67 double_t rhi, rlo;
68 rhi = asdouble (asuint64 (r) & -1ULL << 32);
69 rlo = r - rhi;
70 hi = rhi * InvLn2hi;
71 lo = rlo * InvLn2hi + r * InvLn2lo;
72 #endif
73 r2 = r * r; /* rounding error: 0x1p-62. */
74 r4 = r2 * r2;
75 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
76 p = r2 * (B[0] + r * B[1]);
77 y = hi + p;
78 lo += hi - y + p;
79 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
80 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
81 y += lo;
82 return y;
83 }
84 if (__glibc_unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
85 {
86 /* x < 0x1p-1022 or inf or nan. */
87 if (ix * 2 == 0)
88 return __math_divzero (1);
89 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
90 return x;
91 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
92 return __math_invalid (x);
93 /* x is subnormal, normalize it. */
94 ix = asuint64 (x * 0x1p52);
95 ix -= 52ULL << 52;
96 }
97
98 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
99 The range is split into N subintervals.
100 The ith subinterval contains z and c is near its center. */
101 tmp = ix - OFF;
102 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
103 k = (int64_t) tmp >> 52; /* arithmetic shift */
104 iz = ix - (tmp & 0xfffULL << 52);
105 invc = T[i].invc;
106 logc = T[i].logc;
107 z = asdouble (iz);
108 kd = (double_t) k;
109
110 /* log2(x) = log2(z/c) + log2(c) + k. */
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 t1 = r * InvLn2hi;
116 t2 = r * InvLn2lo + __builtin_fma (r, InvLn2hi, -t1);
117 #else
118 double_t rhi, rlo;
119 /* rounding error: 0x1p-55/N + 0x1p-65. */
120 r = (z - T2[i].chi - T2[i].clo) * invc;
121 rhi = asdouble (asuint64 (r) & -1ULL << 32);
122 rlo = r - rhi;
123 t1 = rhi * InvLn2hi;
124 t2 = rlo * InvLn2hi + r * InvLn2lo;
125 #endif
126
127 /* hi + lo = r/ln2 + log2(c) + k. */
128 t3 = kd + logc;
129 hi = t3 + t1;
130 lo = t3 - hi + t1 + t2;
131
132 /* log2(r+1) = r/ln2 + r^2*poly(r). */
133 /* Evaluation is optimized assuming superscalar pipelined execution. */
134 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
135 r4 = r2 * r2;
136 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
137 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
138 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
139 y = lo + r2 * p + hi;
140 return y;
141 }
142 #ifndef __log2
143 strong_alias (__log2, __ieee754_log2)
144 libm_alias_finite (__ieee754_log2, __log2)
145 # if LIBM_SVID_COMPAT
146 versioned_symbol (libm, __log2, log2, GLIBC_2_29);
147 libm_alias_double_other (__log2, log2)
148 # else
149 libm_alias_double (__log2, log2)
150 # endif
151 #endif
152