1 /*
2 * Double-precision log(x) function.
3 *
4 * Copyright (c) 2018-2024, 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 #include "test_defs.h"
13 #include "test_sig.h"
14
15 #define T __log_data.tab
16 #define T2 __log_data.tab2
17 #define B __log_data.poly1
18 #define A __log_data.poly
19 #define Ln2hi __log_data.ln2hi
20 #define Ln2lo __log_data.ln2lo
21 #define N (1 << LOG_TABLE_BITS)
22 #define OFF 0x3fe6000000000000
23
24 /* Top 16 bits of a double. */
25 static inline uint32_t
top16(double x)26 top16 (double x)
27 {
28 return asuint64 (x) >> 48;
29 }
30
31 double
log(double x)32 log (double x)
33 {
34 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
35 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
36 uint64_t ix, iz, tmp;
37 uint32_t top;
38 int k, i;
39
40 ix = asuint64 (x);
41 top = top16 (x);
42
43 #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
44 # define LO asuint64 (1.0 - 0x1p-5)
45 # define HI asuint64 (1.0 + 0x1.1p-5)
46 #elif LOG_POLY1_ORDER == 12
47 # define LO asuint64 (1.0 - 0x1p-4)
48 # define HI asuint64 (1.0 + 0x1.09p-4)
49 #endif
50 if (unlikely (ix - LO < HI - LO))
51 {
52 /* Handle close to 1.0 inputs separately. */
53 /* Fix sign of zero with downward rounding when x==1. */
54 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
55 return 0;
56 r = x - 1.0;
57 r2 = r * r;
58 r3 = r * r2;
59 #if LOG_POLY1_ORDER == 10
60 /* Worst-case error is around 0.516 ULP. */
61 y = r3 * (B[1] + r * B[2] + r2 * B[3]
62 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
63 w = B[0] * r2; /* B[0] == -0.5. */
64 hi = r + w;
65 y += r - hi + w;
66 y += hi;
67 #elif LOG_POLY1_ORDER == 11
68 /* Worst-case error is around 0.516 ULP. */
69 y = r3 * (B[1] + r * B[2]
70 + r2 * (B[3] + r * B[4] + r2 * B[5]
71 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
72 w = B[0] * r2; /* B[0] == -0.5. */
73 hi = r + w;
74 y += r - hi + w;
75 y += hi;
76 #elif LOG_POLY1_ORDER == 12
77 y = r3 * (B[1] + r * B[2] + r2 * B[3]
78 + r3 * (B[4] + r * B[5] + r2 * B[6]
79 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
80 # if N <= 64
81 /* Worst-case error is around 0.532 ULP. */
82 w = B[0] * r2; /* B[0] == -0.5. */
83 hi = r + w;
84 y += r - hi + w;
85 y += hi;
86 # else
87 /* Worst-case error is around 0.507 ULP. */
88 w = r * 0x1p27;
89 double_t rhi = r + w - w;
90 double_t rlo = r - rhi;
91 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
92 hi = r + w;
93 lo = r - hi + w;
94 lo += B[0] * rlo * (rhi + r);
95 y += lo;
96 y += hi;
97 # endif
98 #endif
99 return eval_as_double (y);
100 }
101 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
102 {
103 /* x < 0x1p-1022 or inf or nan. */
104 if (ix * 2 == 0)
105 return __math_divzero (1);
106 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
107 return x;
108 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
109 return __math_invalid (x);
110 /* x is subnormal, normalize it. */
111 ix = asuint64 (x * 0x1p52);
112 ix -= 52ULL << 52;
113 }
114
115 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
116 The range is split into N subintervals.
117 The ith subinterval contains z and c is near its center. */
118 tmp = ix - OFF;
119 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
120 k = (int64_t) tmp >> 52; /* arithmetic shift */
121 iz = ix - (tmp & 0xfffULL << 52);
122 invc = T[i].invc;
123 logc = T[i].logc;
124 z = asdouble (iz);
125
126 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
127 /* r ~= z/c - 1, |r| < 1/(2*N). */
128 #if HAVE_FAST_FMA
129 /* rounding error: 0x1p-55/N. */
130 r = fma (z, invc, -1.0);
131 #else
132 /* rounding error: 0x1p-55/N + 0x1p-66. */
133 r = (z - T2[i].chi - T2[i].clo) * invc;
134 #endif
135 kd = (double_t) k;
136
137 /* hi + lo = r + log(c) + k*Ln2. */
138 w = kd * Ln2hi + logc;
139 hi = w + r;
140 lo = w - hi + r + kd * Ln2lo;
141
142 /* log(x) = lo + (log1p(r) - r) + hi. */
143 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
144 /* Worst case error if |y| > 0x1p-5:
145 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
146 Worst case error if |y| > 0x1p-4:
147 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
148 #if LOG_POLY_ORDER == 6
149 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
150 #elif LOG_POLY_ORDER == 7
151 y = lo
152 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
153 + r2 * r2 * (A[4] + r * A[5]))
154 + hi;
155 #endif
156 return eval_as_double (y);
157 }
158 #if USE_GLIBC_ABI
strong_alias(log,__log_finite)159 strong_alias (log, __log_finite)
160 hidden_alias (log, __ieee754_log)
161 # if LDBL_MANT_DIG == 53
162 long double logl (long double x) { return log (x); }
163 # endif
164 #endif
165
166 TEST_SIG (S, D, 1, log, 0.01, 11.1)
167 TEST_ULP (log, 0.02)
168 TEST_ULP_NONNEAREST (log, 0.5)
169 TEST_INTERVAL (log, 0, 0xffff000000000000, 10000)
170 TEST_INTERVAL (log, 0x1p-4, 0x1p4, 400000)
171 TEST_INTERVAL (log, 0, inf, 400000)
172