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-exception
7 */
8
9 #include <float.h>
10 #include <math.h>
11 #include <stdint.h>
12 #include "math_config.h"
13
14 #define T __log_data.tab
15 #define T2 __log_data.tab2
16 #define B __log_data.poly1
17 #define A __log_data.poly
18 #define Ln2hi __log_data.ln2hi
19 #define Ln2lo __log_data.ln2lo
20 #define N (1 << LOG_TABLE_BITS)
21 #define OFF 0x3fe6000000000000
22
23 /* Top 16 bits of a double. */
24 static inline uint32_t
top16(double x)25 top16 (double x)
26 {
27 return asuint64 (x) >> 48;
28 }
29
30 double
log(double x)31 log (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 == 11
43 # define LO asuint64 (1.0 - 0x1p-5)
44 # define HI asuint64 (1.0 + 0x1.1p-5)
45 #elif LOG_POLY1_ORDER == 12
46 # define LO asuint64 (1.0 - 0x1p-4)
47 # define HI asuint64 (1.0 + 0x1.09p-4)
48 #endif
49 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 == 10
59 /* 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 == 11
67 /* 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 == 12
76 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 <= 64
80 /* 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 # else
86 /* 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 # endif
97 #endif
98 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_FMA
128 /* rounding error: 0x1p-55/N. */
129 r = fma (z, invc, -1.0);
130 #else
131 /* rounding error: 0x1p-55/N + 0x1p-66. */
132 r = (z - T2[i].chi - T2[i].clo) * invc;
133 #endif
134 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 == 6
148 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
149 #elif LOG_POLY_ORDER == 7
150 y = lo
151 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
152 + r2 * r2 * (A[4] + r * A[5]))
153 + hi;
154 #endif
155 return eval_as_double (y);
156 }
157 #if USE_GLIBC_ABI
strong_alias(log,__log_finite)158 strong_alias (log, __log_finite)
159 hidden_alias (log, __ieee754_log)
160 # if LDBL_MANT_DIG == 53
161 long double logl (long double x) { return log (x); }
162 # endif
163 #endif
164