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320054e8 | 1 | /* |
d4db3fa2 | 2 | * Double-precision log(x) function. |
320054e8 | 3 | * |
d4db3fa2 DG |
4 | * Copyright (c) 2018, Arm Limited. |
5 | * SPDX-License-Identifier: MIT | |
320054e8 DG |
6 | */ |
7 | ||
8 | #include <math.h> | |
9 | #include <stdint.h> | |
d4db3fa2 DG |
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 | |
320054e8 | 21 | |
d4db3fa2 DG |
22 | /* Top 16 bits of a double. */ |
23 | static inline uint32_t top16(double x) | |
24 | { | |
25 | return asuint64(x) >> 48; | |
26 | } | |
320054e8 DG |
27 | |
28 | double log(double x) | |
29 | { | |
d4db3fa2 DG |
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); | |
320054e8 | 86 | |
d4db3fa2 DG |
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; | |
320054e8 | 97 | |
d4db3fa2 DG |
98 | /* hi + lo = r + log(c) + k*Ln2. */ |
99 | w = kd * Ln2hi + logc; | |
100 | hi = w + r; | |
101 | lo = w - hi + r + kd * Ln2lo; | |
320054e8 | 102 | |
d4db3fa2 DG |
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); | |
320054e8 | 112 | } |