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 #ifdef NEED_MATH_DIVZERO
23 /* base math internal func */
__math_divzero(uint32_t sign)24 double __math_divzero(uint32_t sign)
25 {
26 return fp_barrier(sign ? -1.0 : 1.0) / 0.0;
27 }
28 #endif
29
30 #ifdef NEED_MATH_INVALID
__math_invalid(double x)31 double __math_invalid(double x)
32 {
33 return (x - x) / (x - x);
34 }
35 #endif
36
37 /* Top 16 bits of a double. */
top16(double x)38 static inline uint32_t top16(double x)
39 {
40 return asuint64(x) >> 48;
41 }
42
log(double x)43 double log(double x)
44 {
45 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
46 uint64_t ix, iz, tmp;
47 uint32_t top;
48 int k, i;
49
50 ix = asuint64(x);
51 top = top16(x);
52 #define LO asuint64(1.0 - 0x1p-4)
53 #define HI asuint64(1.0 + 0x1.09p-4)
54 if (predict_false(ix - LO < HI - LO)) {
55 /* Handle close to 1.0 inputs separately. */
56 /* Fix sign of zero with downward rounding when x==1. */
57 if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
58 return 0;
59 r = x - 1.0;
60 r2 = r * r;
61 r3 = r * r2;
62 y = r3 *
63 (B[1] + r * B[2] + r2 * B[3] +
64 r3 * (B[4] + r * B[5] + r2 * B[6] +
65 r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
66 /* Worst-case error is around 0.507 ULP. */
67 w = r * 0x1p27;
68 double_t rhi = r + w - w;
69 double_t rlo = r - rhi;
70 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
71 hi = r + w;
72 lo = r - hi + w;
73 lo += B[0] * rlo * (rhi + r);
74 y += lo;
75 y += hi;
76 return eval_as_double(y);
77 }
78 if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
79 /* x < 0x1p-1022 or inf or nan. */
80 if (ix * 2 == 0)
81 return __math_divzero(1);
82 if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
83 return x;
84 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
85 return __math_invalid(x);
86 /* x is subnormal, normalize it. */
87 ix = asuint64(x * 0x1p52);
88 ix -= 52ULL << 52;
89 }
90
91 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
92 The range is split into N subintervals.
93 The ith subinterval contains z and c is near its center. */
94 tmp = ix - OFF;
95 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
96 k = (int64_t)tmp >> 52; /* arithmetic shift */
97 iz = ix - (tmp & 0xfffULL << 52);
98 invc = T[i].invc;
99 logc = T[i].logc;
100 z = asdouble(iz);
101
102 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
103 /* r ~= z/c - 1, |r| < 1/(2*N). */
104 #if __FP_FAST_FMA
105 /* rounding error: 0x1p-55/N. */
106 r = __builtin_fma(z, invc, -1.0);
107 #else
108 /* rounding error: 0x1p-55/N + 0x1p-66. */
109 r = (z - T2[i].chi - T2[i].clo) * invc;
110 #endif
111 kd = (double_t)k;
112
113 /* hi + lo = r + log(c) + k*Ln2. */
114 w = kd * Ln2hi + logc;
115 hi = w + r;
116 lo = w - hi + r + kd * Ln2lo;
117
118 /* log(x) = lo + (log1p(r) - r) + hi. */
119 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
120 /* Worst case error if |y| > 0x1p-5:
121 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
122 Worst case error if |y| > 0x1p-4:
123 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
124 y = lo + r2 * A[0] +
125 r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
126 return eval_as_double(y);
127 }
128