1 /*
2 * Double-precision log(x) function.
3 *
4 * Copyright (c) 2018-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include <float.h>
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.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 /* 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
optr_aor_log_f64(double x)30 optr_aor_log_f64 (double x)
31 {
32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
33 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
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 LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
42 #define LO asuint64 (1.0 - 0x1p-5)
43 #define HI asuint64 (1.0 + 0x1.1p-5)
44 #elif LOG_POLY1_ORDER == 12
45 #define LO asuint64 (1.0 - 0x1p-4)
46 #define HI asuint64 (1.0 + 0x1.09p-4)
47 #endif
48 if (unlikely (ix - LO < HI - LO))
49 {
50 /* Handle close to 1.0 inputs separately. */
51 /* Fix sign of zero with downward rounding when x==1. */
52 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
53 return 0;
54 r = x - 1.0;
55 r2 = r * r;
56 r3 = r * r2;
57 #if LOG_POLY1_ORDER == 10
58 /* Worst-case error is around 0.516 ULP. */
59 y = r3
60 * (B[1] + r * B[2] + r2 * B[3]
61 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
62 w = B[0] * r2; /* B[0] == -0.5. */
63 hi = r + w;
64 y += r - hi + w;
65 y += hi;
66 #elif LOG_POLY1_ORDER == 11
67 /* Worst-case error is around 0.516 ULP. */
68 y = r3
69 * (B[1] + r * B[2]
70 + r2
71 * (B[3] + r * B[4] + r2 * B[5]
72 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
73 w = B[0] * r2; /* B[0] == -0.5. */
74 hi = r + w;
75 y += r - hi + w;
76 y += hi;
77 #elif LOG_POLY1_ORDER == 12
78 y = r3
79 * (B[1] + r * B[2] + r2 * B[3]
80 + r3
81 * (B[4] + r * B[5] + r2 * B[6]
82 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
83 #if N <= 64
84 /* Worst-case error is around 0.532 ULP. */
85 w = B[0] * r2; /* B[0] == -0.5. */
86 hi = r + w;
87 y += r - hi + w;
88 y += hi;
89 #else
90 /* Worst-case error is around 0.507 ULP. */
91 w = r * 0x1p27;
92 double_t rhi = r + w - w;
93 double_t rlo = r - rhi;
94 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
95 hi = r + w;
96 lo = r - hi + w;
97 lo += B[0] * rlo * (rhi + r);
98 y += lo;
99 y += hi;
100 #endif
101 #endif
102 return eval_as_double (y);
103 }
104 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
105 {
106 /* x < 0x1p-1022 or inf or nan. */
107 if (ix * 2 == 0)
108 return __math_divzero (1);
109 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
110 return x;
111 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
112 return __math_invalid (x);
113 /* x is subnormal, normalize it. */
114 ix = asuint64 (x * 0x1p52);
115 ix -= 52ULL << 52;
116 }
117
118 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
119 The range is split into N subintervals.
120 The ith subinterval contains z and c is near its center. */
121 tmp = ix - OFF;
122 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
123 k = (int64_t) tmp >> 52; /* arithmetic shift */
124 iz = ix - (tmp & 0xfffULL << 52);
125 invc = T[i].invc;
126 logc = T[i].logc;
127 z = asdouble (iz);
128
129 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
130 /* r ~= z/c - 1, |r| < 1/(2*N). */
131 #if HAVE_FAST_FMA
132 /* rounding error: 0x1p-55/N. */
133 r = fma (z, invc, -1.0);
134 #else
135 /* rounding error: 0x1p-55/N + 0x1p-66. */
136 r = (z - T2[i].chi - T2[i].clo) * invc;
137 #endif
138 kd = (double_t) k;
139
140 /* hi + lo = r + log(c) + k*Ln2. */
141 w = kd * Ln2hi + logc;
142 hi = w + r;
143 lo = w - hi + r + kd * Ln2lo;
144
145 /* log(x) = lo + (log1p(r) - r) + hi. */
146 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
147 /* Worst case error if |y| > 0x1p-5:
148 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
149 Worst case error if |y| > 0x1p-4:
150 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
151 #if LOG_POLY_ORDER == 6
152 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
153 #elif LOG_POLY_ORDER == 7
154 y = lo
155 + r2
156 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
157 + r2 * r2 * (A[4] + r * A[5]))
158 + hi;
159 #endif
160 return eval_as_double (y);
161 }
162