1 /*
2 * Double-precision log2(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 __log2_data.tab
15 #define T2 __log2_data.tab2
16 #define B __log2_data.poly1
17 #define A __log2_data.poly
18 #define InvLn2hi __log2_data.invln2hi
19 #define InvLn2lo __log2_data.invln2lo
20 #define N (1 << LOG2_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
log2(double x)31 log2 (double x)
32 {
33 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
34 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
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 LOG2_POLY1_ORDER == 11
43 # define LO asuint64 (1.0 - 0x1.5b51p-5)
44 # define HI asuint64 (1.0 + 0x1.6ab2p-5)
45 #endif
46 if (unlikely (ix - LO < HI - LO))
47 {
48 /* Handle close to 1.0 inputs separately. */
49 /* Fix sign of zero with downward rounding when x==1. */
50 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
51 return 0;
52 r = x - 1.0;
53 #if HAVE_FAST_FMA
54 hi = r * InvLn2hi;
55 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);
56 #else
57 double_t rhi, rlo;
58 rhi = asdouble (asuint64 (r) & -1ULL << 32);
59 rlo = r - rhi;
60 hi = rhi * InvLn2hi;
61 lo = rlo * InvLn2hi + r * InvLn2lo;
62 #endif
63 r2 = r * r; /* rounding error: 0x1p-62. */
64 r4 = r2 * r2;
65 #if LOG2_POLY1_ORDER == 11
66 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
67 p = r2 * (B[0] + r * B[1]);
68 y = hi + p;
69 lo += hi - y + p;
70 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
71 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
72 y += lo;
73 #endif
74 return eval_as_double (y);
75 }
76 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
77 {
78 /* x < 0x1p-1022 or inf or nan. */
79 if (ix * 2 == 0)
80 return __math_divzero (1);
81 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
82 return x;
83 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
84 return __math_invalid (x);
85 /* x is subnormal, normalize it. */
86 ix = asuint64 (x * 0x1p52);
87 ix -= 52ULL << 52;
88 }
89
90 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
91 The range is split into N subintervals.
92 The ith subinterval contains z and c is near its center. */
93 tmp = ix - OFF;
94 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
95 k = (int64_t) tmp >> 52; /* arithmetic shift */
96 iz = ix - (tmp & 0xfffULL << 52);
97 invc = T[i].invc;
98 logc = T[i].logc;
99 z = asdouble (iz);
100 kd = (double_t) k;
101
102 /* log2(x) = log2(z/c) + log2(c) + k. */
103 /* r ~= z/c - 1, |r| < 1/(2*N). */
104 #if HAVE_FAST_FMA
105 /* rounding error: 0x1p-55/N. */
106 r = fma (z, invc, -1.0);
107 t1 = r * InvLn2hi;
108 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);
109 #else
110 double_t rhi, rlo;
111 /* rounding error: 0x1p-55/N + 0x1p-65. */
112 r = (z - T2[i].chi - T2[i].clo) * invc;
113 rhi = asdouble (asuint64 (r) & -1ULL << 32);
114 rlo = r - rhi;
115 t1 = rhi * InvLn2hi;
116 t2 = rlo * InvLn2hi + r * InvLn2lo;
117 #endif
118
119 /* hi + lo = r/ln2 + log2(c) + k. */
120 t3 = kd + logc;
121 hi = t3 + t1;
122 lo = t3 - hi + t1 + t2;
123
124 /* log2(r+1) = r/ln2 + r^2*poly(r). */
125 /* Evaluation is optimized assuming superscalar pipelined execution. */
126 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
127 r4 = r2 * r2;
128 #if LOG2_POLY_ORDER == 7
129 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
130 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
131 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
132 y = lo + r2 * p + hi;
133 #endif
134 return eval_as_double (y);
135 }
136 #if USE_GLIBC_ABI
strong_alias(log2,__log2_finite)137 strong_alias (log2, __log2_finite)
138 hidden_alias (log2, __ieee754_log2)
139 # if LDBL_MANT_DIG == 53
140 long double log2l (long double x) { return log2 (x); }
141 # endif
142 #endif
143