1/*
2 * Double-precision log2(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 "log2_data.h"
12
13#define T __log2_data.tab
14#define T2 __log2_data.tab2
15#define B __log2_data.poly1
16#define A __log2_data.poly
17#define InvLn2hi __log2_data.invln2hi
18#define InvLn2lo __log2_data.invln2lo
19#define N (1 << LOG2_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 log2(double x)
29{
30 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
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 - 0x1.5b51p-5)
38#define HI asuint64(1.0 + 0x1.6ab2p-5)
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#if __FP_FAST_FMA
46 hi = r * InvLn2hi;
47 lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
48#else
49 double_t rhi, rlo;
50 rhi = asdouble(asuint64(r) & -1ULL << 32);
51 rlo = r - rhi;
52 hi = rhi * InvLn2hi;
53 lo = rlo * InvLn2hi + r * InvLn2lo;
54#endif
55 r2 = r * r; /* rounding error: 0x1p-62. */
56 r4 = r2 * r2;
57 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
58 p = r2 * (B[0] + r * B[1]);
59 y = hi + p;
60 lo += hi - y + p;
61 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
62 r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
63 y += lo;
64 return eval_as_double(y);
65 }
66 if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
67 /* x < 0x1p-1022 or inf or nan. */
68 if (ix * 2 == 0)
69 return __math_divzero(1);
70 if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
71 return x;
72 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
73 return __math_invalid(x);
74 /* x is subnormal, normalize it. */
75 ix = asuint64(x * 0x1p52);
76 ix -= 52ULL << 52;
77 }
78
79 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
80 The range is split into N subintervals.
81 The ith subinterval contains z and c is near its center. */
82 tmp = ix - OFF;
83 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
84 k = (int64_t)tmp >> 52; /* arithmetic shift */
85 iz = ix - (tmp & 0xfffULL << 52);
86 invc = T[i].invc;
87 logc = T[i].logc;
88 z = asdouble(iz);
89 kd = (double_t)k;
90
91 /* log2(x) = log2(z/c) + log2(c) + k. */
92 /* r ~= z/c - 1, |r| < 1/(2*N). */
93#if __FP_FAST_FMA
94 /* rounding error: 0x1p-55/N. */
95 r = __builtin_fma(z, invc, -1.0);
96 t1 = r * InvLn2hi;
97 t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
98#else
99 double_t rhi, rlo;
100 /* rounding error: 0x1p-55/N + 0x1p-65. */
101 r = (z - T2[i].chi - T2[i].clo) * invc;
102 rhi = asdouble(asuint64(r) & -1ULL << 32);
103 rlo = r - rhi;
104 t1 = rhi * InvLn2hi;
105 t2 = rlo * InvLn2hi + r * InvLn2lo;
106#endif
107
108 /* hi + lo = r/ln2 + log2(c) + k. */
109 t3 = kd + logc;
110 hi = t3 + t1;
111 lo = t3 - hi + t1 + t2;
112
113 /* log2(r+1) = r/ln2 + r^2*poly(r). */
114 /* Evaluation is optimized assuming superscalar pipelined execution. */
115 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
116 r4 = r2 * r2;
117 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
118 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
119 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
120 y = lo + r2 * p + hi;
121 return eval_as_double(y);
122}
123