1 /* Double-precision log(x) function.
2 Copyright (c) 2018 Arm Ltd. All rights reserved.
3
4 SPDX-License-Identifier: BSD-3-Clause
5
6 Redistribution and use in source and binary forms, with or without
7 modification, are permitted provided that the following conditions
8 are met:
9 1. Redistributions of source code must retain the above copyright
10 notice, this list of conditions and the following disclaimer.
11 2. Redistributions in binary form must reproduce the above copyright
12 notice, this list of conditions and the following disclaimer in the
13 documentation and/or other materials provided with the distribution.
14 3. The name of the company may not be used to endorse or promote
15 products derived from this software without specific prior written
16 permission.
17
18 THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED
19 WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
20 MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
21 IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
22 SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
23 TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
24 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
25 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
26 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
27 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
28
29 #include "fdlibm.h"
30 #if !__OBSOLETE_MATH_DOUBLE
31
32 #include <math.h>
33 #include <stdint.h>
34 #include "math_config.h"
35
36 #define T __log_data.tab
37 #define T2 __log_data.tab2
38 #define B __log_data.poly1
39 #define A __log_data.poly
40 #define Ln2hi __log_data.ln2hi
41 #define Ln2lo __log_data.ln2lo
42 #define N (1 << LOG_TABLE_BITS)
43 #define OFF 0x3fe6000000000000
44
45 /* Top 16 bits of a double. */
46 static inline uint32_t
top16(double x)47 top16 (double x)
48 {
49 return asuint64 (x) >> 48;
50 }
51
52 double
log(double x)53 log (double x)
54 {
55 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
56 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
57 uint64_t ix, iz, tmp;
58 uint32_t top;
59 int k, i;
60
61 ix = asuint64 (x);
62 top = top16 (x);
63
64 #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
65 # define LO asuint64 (1.0 - 0x1p-5)
66 # define HI asuint64 (1.0 + 0x1.1p-5)
67 #elif LOG_POLY1_ORDER == 12
68 # define LO asuint64 (1.0 - 0x1p-4)
69 # define HI asuint64 (1.0 + 0x1.09p-4)
70 #endif
71 if (unlikely (ix - LO < HI - LO))
72 {
73 /* Handle close to 1.0 inputs separately. */
74 /* Fix sign of zero with downward rounding when x==1. */
75 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
76 return 0;
77 r = x - 1.0;
78 r2 = r * r;
79 r3 = r * r2;
80 #if LOG_POLY1_ORDER == 10
81 /* Worst-case error is around 0.516 ULP. */
82 y = r3 * (B[1] + r * B[2] + r2 * B[3]
83 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
84 w = B[0] * r2; /* B[0] == -0.5. */
85 hi = r + w;
86 y += r - hi + w;
87 y += hi;
88 #elif LOG_POLY1_ORDER == 11
89 /* Worst-case error is around 0.516 ULP. */
90 y = r3 * (B[1] + r * B[2]
91 + r2 * (B[3] + r * B[4] + r2 * B[5]
92 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
93 w = B[0] * r2; /* B[0] == -0.5. */
94 hi = r + w;
95 y += r - hi + w;
96 y += hi;
97 #elif LOG_POLY1_ORDER == 12
98 y = r3 * (B[1] + r * B[2] + r2 * B[3]
99 + r3 * (B[4] + r * B[5] + r2 * B[6]
100 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
101 # if N <= 64
102 /* Worst-case error is around 0.532 ULP. */
103 w = B[0] * r2; /* B[0] == -0.5. */
104 hi = r + w;
105 y += r - hi + w;
106 y += hi;
107 # else
108 /* Worst-case error is around 0.507 ULP. */
109 w = r * 0x1p27;
110 double_t rhi = r + w - w;
111 double_t rlo = r - rhi;
112 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
113 hi = r + w;
114 lo = r - hi + w;
115 lo += B[0] * rlo * (rhi + r);
116 y += lo;
117 y += hi;
118 # endif
119 #endif
120 return y;
121 }
122 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
123 {
124 /* x < 0x1p-1022 or inf or nan. */
125 if (ix * 2 == 0)
126 return __math_divzero (1);
127 if (ix == asuint64 ((double) INFINITY)) /* log(inf) == inf. */
128 return x;
129 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
130 return __math_invalid (x);
131 /* x is subnormal, normalize it. */
132 ix = asuint64 (x * 0x1p52);
133 ix -= 52ULL << 52;
134 }
135
136 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
137 The range is split into N subintervals.
138 The ith subinterval contains z and c is near its center. */
139 tmp = ix - OFF;
140 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
141 k = (int64_t) tmp >> 52; /* arithmetic shift */
142 iz = ix - (tmp & 0xfffULL << 52);
143 invc = T[i].invc;
144 logc = T[i].logc;
145 z = asfloat64 (iz);
146
147 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
148 /* r ~= z/c - 1, |r| < 1/(2*N). */
149 #if _HAVE_FAST_FMA
150 /* rounding error: 0x1p-55/N. */
151 r = fma (z, invc, -1.0);
152 #else
153 /* rounding error: 0x1p-55/N + 0x1p-66. */
154 r = (z - T2[i].chi - T2[i].clo) * invc;
155 #endif
156 kd = (double_t) k;
157
158 /* hi + lo = r + log(c) + k*Ln2. */
159 w = kd * Ln2hi + logc;
160 hi = w + r;
161 lo = w - hi + r + kd * Ln2lo;
162
163 /* log(x) = lo + (log1p(r) - r) + hi. */
164 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
165 /* Worst case error if |y| > 0x1p-5:
166 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
167 Worst case error if |y| > 0x1p-4:
168 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
169 #if LOG_POLY_ORDER == 6
170 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
171 #elif LOG_POLY_ORDER == 7
172 y = lo
173 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
174 + r2 * r2 * (A[4] + r * A[5]))
175 + hi;
176 #endif
177 return y;
178 }
179
180 _MATH_ALIAS_d_d(log)
181
182 #endif
183
