1/*
2 * Double-precision log(x) function.
3 *
4 * Copyright (c) 2018, Arm Limited.
5 * SPDX-License-Identifier: MIT
6 */
7
8#include <math.h>
9#include <stdint.h>
10#include "libm.h"
11#include "log_data.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 top16(double x)
24{
25 return asuint64(x) >> 48;
26}
27
28double log(double x)
29{
30 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
31 uint64_t ix, iz, tmp;
32 uint32_t top;
33 int k, i;
34
35 ix = asuint64(x);
36 top = top16(x);
37#define LO asuint64(1.0 - 0x1p-4)
38#define HI asuint64(1.0 + 0x1.09p-4)
39 if (predict_false(ix - LO < HI - LO)) {
40 /* Handle close to 1.0 inputs separately. */
41 /* Fix sign of zero with downward rounding when x==1. */
42 if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
43 return 0;
44 r = x - 1.0;
45 r2 = r * r;
46 r3 = r * r2;
47 y = r3 *
48 (B[1] + r * B[2] + r2 * B[3] +
49 r3 * (B[4] + r * B[5] + r2 * B[6] +
50 r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
51 /* Worst-case error is around 0.507 ULP. */
52 w = r * 0x1p27;
53 double_t rhi = r + w - w;
54 double_t rlo = r - rhi;
55 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
56 hi = r + w;
57 lo = r - hi + w;
58 lo += B[0] * rlo * (rhi + r);
59 y += lo;
60 y += hi;
61 return eval_as_double(y);
62 }
63 if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
64 /* x < 0x1p-1022 or inf or nan. */
65 if (ix * 2 == 0)
66 return __math_divzero(1);
67 if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
68 return x;
69 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
70 return __math_invalid(x);
71 /* x is subnormal, normalize it. */
72 ix = asuint64(x * 0x1p52);
73 ix -= 52ULL << 52;
74 }
75
76 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
77 The range is split into N subintervals.
78 The ith subinterval contains z and c is near its center. */
79 tmp = ix - OFF;
80 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
81 k = (int64_t)tmp >> 52; /* arithmetic shift */
82 iz = ix - (tmp & 0xfffULL << 52);
83 invc = T[i].invc;
84 logc = T[i].logc;
85 z = asdouble(iz);
86
87 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
88 /* r ~= z/c - 1, |r| < 1/(2*N). */
89#if __FP_FAST_FMA
90 /* rounding error: 0x1p-55/N. */
91 r = __builtin_fma(z, invc, -1.0);
92#else
93 /* rounding error: 0x1p-55/N + 0x1p-66. */
94 r = (z - T2[i].chi - T2[i].clo) * invc;
95#endif
96 kd = (double_t)k;
97
98 /* hi + lo = r + log(c) + k*Ln2. */
99 w = kd * Ln2hi + logc;
100 hi = w + r;
101 lo = w - hi + r + kd * Ln2lo;
102
103 /* log(x) = lo + (log1p(r) - r) + hi. */
104 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
105 /* Worst case error if |y| > 0x1p-5:
106 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
107 Worst case error if |y| > 0x1p-4:
108 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
109 y = lo + r2 * A[0] +
110 r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
111 return eval_as_double(y);
112}
113