1 /*
2 * Double-precision log2(x) function.
3 *
4 * Copyright (c) 2018-2019, Arm Limited.
5 * SPDX-License-Identifier: MIT
6 */
7
8 #include <float.h>
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.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. */
23 static inline uint32_t
top16(double x)24 top16 (double x)
25 {
26 return asuint64 (x) >> 48;
27 }
28
29 double
log2(double x)30 log2 (double x)
31 {
32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
33 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
34 uint64_t ix, iz, tmp;
35 uint32_t top;
36 int k, i;
37
38 ix = asuint64 (x);
39 top = top16 (x);
40
41 #if LOG2_POLY1_ORDER == 11
42 # define LO asuint64 (1.0 - 0x1.5b51p-5)
43 # define HI asuint64 (1.0 + 0x1.6ab2p-5)
44 #endif
45 if (unlikely (ix - LO < HI - LO))
46 {
47 /* Handle close to 1.0 inputs separately. */
48 /* Fix sign of zero with downward rounding when x==1. */
49 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
50 return 0;
51 r = x - 1.0;
52 #if HAVE_FAST_FMA
53 hi = r * InvLn2hi;
54 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);
55 #else
56 double_t rhi, rlo;
57 rhi = asdouble (asuint64 (r) & -1ULL << 32);
58 rlo = r - rhi;
59 hi = rhi * InvLn2hi;
60 lo = rlo * InvLn2hi + r * InvLn2lo;
61 #endif
62 r2 = r * r; /* rounding error: 0x1p-62. */
63 r4 = r2 * r2;
64 #if LOG2_POLY1_ORDER == 11
65 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
66 p = r2 * (B[0] + r * B[1]);
67 y = hi + p;
68 lo += hi - y + p;
69 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
70 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
71 y += lo;
72 #endif
73 return eval_as_double (y);
74 }
75 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
76 {
77 /* x < 0x1p-1022 or inf or nan. */
78 if (ix * 2 == 0)
79 return __math_divzero (1);
80 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
81 return x;
82 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
83 return __math_invalid (x);
84 /* x is subnormal, normalize it. */
85 ix = asuint64 (x * 0x1p52);
86 ix -= 52ULL << 52;
87 }
88
89 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
90 The range is split into N subintervals.
91 The ith subinterval contains z and c is near its center. */
92 tmp = ix - OFF;
93 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
94 k = (int64_t) tmp >> 52; /* arithmetic shift */
95 iz = ix - (tmp & 0xfffULL << 52);
96 invc = T[i].invc;
97 logc = T[i].logc;
98 z = asdouble (iz);
99 kd = (double_t) k;
100
101 /* log2(x) = log2(z/c) + log2(c) + k. */
102 /* r ~= z/c - 1, |r| < 1/(2*N). */
103 #if HAVE_FAST_FMA
104 /* rounding error: 0x1p-55/N. */
105 r = fma (z, invc, -1.0);
106 t1 = r * InvLn2hi;
107 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);
108 #else
109 double_t rhi, rlo;
110 /* rounding error: 0x1p-55/N + 0x1p-65. */
111 r = (z - T2[i].chi - T2[i].clo) * invc;
112 rhi = asdouble (asuint64 (r) & -1ULL << 32);
113 rlo = r - rhi;
114 t1 = rhi * InvLn2hi;
115 t2 = rlo * InvLn2hi + r * InvLn2lo;
116 #endif
117
118 /* hi + lo = r/ln2 + log2(c) + k. */
119 t3 = kd + logc;
120 hi = t3 + t1;
121 lo = t3 - hi + t1 + t2;
122
123 /* log2(r+1) = r/ln2 + r^2*poly(r). */
124 /* Evaluation is optimized assuming superscalar pipelined execution. */
125 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
126 r4 = r2 * r2;
127 #if LOG2_POLY_ORDER == 7
128 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
129 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
130 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
131 y = lo + r2 * p + hi;
132 #endif
133 return eval_as_double (y);
134 }
135 #if USE_GLIBC_ABI
strong_alias(log2,__log2_finite)136 strong_alias (log2, __log2_finite)
137 hidden_alias (log2, __ieee754_log2)
138 # if LDBL_MANT_DIG == 53
139 long double log2l (long double x) { return log2 (x); }
140 # endif
141 #endif
142