1 /*
2 * Double-precision log10(x) function.
3 *
4 * Copyright (c) 2020-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "math_config.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11
12 /* Polynomial coefficients and lookup tables. */
13 #define T __log10_data.tab
14 #define T2 __log10_data.tab2
15 #define B __log10_data.poly1
16 #define A __log10_data.poly
17 #define Ln2hi __log10_data.ln2hi
18 #define Ln2lo __log10_data.ln2lo
19 #define InvLn10 __log10_data.invln10
20 #define N (1 << LOG10_TABLE_BITS)
21 #define OFF 0x3fe6000000000000
22 #define LO asuint64 (1.0 - 0x1p-4)
23 #define HI asuint64 (1.0 + 0x1.09p-4)
24
25 /* Top 16 bits of a double. */
26 static inline uint32_t
top16(double x)27 top16 (double x)
28 {
29 return asuint64 (x) >> 48;
30 }
31
32 /* Fast and low accuracy implementation of log10.
33 The implementation is similar to that of math/log, except that:
34 - Polynomials are computed for log10(1+r) with r on same intervals as log.
35 - Lookup parameters are scaled (at runtime) to switch from base e to base 10.
36 Many errors above 1.59 ulp are observed across the whole range of doubles.
37 The greatest observed error is 1.61 ulp, at around 0.965:
38 log10(0x1.dc8710333a29bp-1) got -0x1.fee26884905a6p-6
39 want -0x1.fee26884905a8p-6. */
40 double
log10(double x)41 log10 (double x)
42 {
43 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
44 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
45 uint64_t ix, iz, tmp;
46 uint32_t top;
47 int k, i;
48
49 ix = asuint64 (x);
50 top = top16 (x);
51
52 if (unlikely (ix - LO < HI - LO))
53 {
54 /* Handle close to 1.0 inputs separately. */
55 /* Fix sign of zero with downward rounding when x==1. */
56 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
57 return 0;
58 r = x - 1.0;
59 r2 = r * r;
60 r3 = r * r2;
61 y = r3
62 * (B[1] + r * B[2] + r2 * B[3]
63 + r3
64 * (B[4] + r * B[5] + r2 * B[6]
65 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
66 /* Worst-case error is around 0.507 ULP. */
67 w = r * 0x1p27;
68 double_t rhi = r + w - w;
69 double_t rlo = r - rhi;
70 w = rhi * rhi * B[0];
71 hi = r + w;
72 lo = r - hi + w;
73 lo += B[0] * rlo * (rhi + r);
74 y += lo;
75 y += hi;
76 /* Scale by 1/ln(10). Polynomial already contains scaling. */
77 y = y * InvLn10;
78
79 return eval_as_double (y);
80 }
81 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
82 {
83 /* x < 0x1p-1022 or inf or nan. */
84 if (ix * 2 == 0)
85 return __math_divzero (1);
86 if (ix == asuint64 (INFINITY)) /* log10(inf) == inf. */
87 return x;
88 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
89 return __math_invalid (x);
90 /* x is subnormal, normalize it. */
91 ix = asuint64 (x * 0x1p52);
92 ix -= 52ULL << 52;
93 }
94
95 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
96 The range is split into N subintervals.
97 The ith subinterval contains z and c is near its center. */
98 tmp = ix - OFF;
99 i = (tmp >> (52 - LOG10_TABLE_BITS)) % N;
100 k = (int64_t) tmp >> 52; /* arithmetic shift. */
101 iz = ix - (tmp & 0xfffULL << 52);
102 invc = T[i].invc;
103 logc = T[i].logc;
104 z = asdouble (iz);
105
106 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
107 /* r ~= z/c - 1, |r| < 1/(2*N). */
108 #if HAVE_FAST_FMA
109 /* rounding error: 0x1p-55/N. */
110 r = fma (z, invc, -1.0);
111 #else
112 /* rounding error: 0x1p-55/N + 0x1p-66. */
113 r = (z - T2[i].chi - T2[i].clo) * invc;
114 #endif
115 kd = (double_t) k;
116
117 /* w = log(c) + k*Ln2hi. */
118 w = kd * Ln2hi + logc;
119 hi = w + r;
120 lo = w - hi + r + kd * Ln2lo;
121
122 /* log10(x) = (w + r)/log(10) + (log10(1+r) - r/log(10)). */
123 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
124
125 /* Scale by 1/ln(10). Polynomial already contains scaling. */
126 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
127 y = y * InvLn10;
128
129 return eval_as_double (y);
130 }
131
132 // clang-format off
133 #if USE_GLIBC_ABI
strong_alias(log10,__log10_finite)134 strong_alias (log10, __log10_finite)
135 hidden_alias (log10, __ieee754_log10)
136 #if LDBL_MANT_DIG == 53
137 long double
138 log10l (long double x)
139 {
140 return log10 (x);
141 }
142 #endif
143 #endif
144 // clang-format on
145
146 PL_SIG (S, D, 1, log10, 0.01, 11.1)
147 PL_TEST_ULP (log10, 1.11)
148 PL_TEST_INTERVAL (log10, 0, 0xffff000000000000, 10000)
149 PL_TEST_INTERVAL (log10, 0x1p-4, 0x1p4, 40000)
150 PL_TEST_INTERVAL (log10, 0, inf, 40000)
151