1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-precision log10(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2020-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner
8*f3087befSAndrew Turner #include "math_config.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner
12*f3087befSAndrew Turner /* Polynomial coefficients and lookup tables. */
13*f3087befSAndrew Turner #define T __log10_data.tab
14*f3087befSAndrew Turner #define T2 __log10_data.tab2
15*f3087befSAndrew Turner #define B __log10_data.poly1
16*f3087befSAndrew Turner #define A __log10_data.poly
17*f3087befSAndrew Turner #define Ln2hi __log10_data.ln2hi
18*f3087befSAndrew Turner #define Ln2lo __log10_data.ln2lo
19*f3087befSAndrew Turner #define InvLn10 __log10_data.invln10
20*f3087befSAndrew Turner #define N (1 << LOG10_TABLE_BITS)
21*f3087befSAndrew Turner #define OFF 0x3fe6000000000000
22*f3087befSAndrew Turner #define LO asuint64 (1.0 - 0x1p-4)
23*f3087befSAndrew Turner #define HI asuint64 (1.0 + 0x1.09p-4)
24*f3087befSAndrew Turner
25*f3087befSAndrew Turner /* Top 16 bits of a double. */
26*f3087befSAndrew Turner static inline uint32_t
top16(double x)27*f3087befSAndrew Turner top16 (double x)
28*f3087befSAndrew Turner {
29*f3087befSAndrew Turner return asuint64 (x) >> 48;
30*f3087befSAndrew Turner }
31*f3087befSAndrew Turner
32*f3087befSAndrew Turner /* Fast and low accuracy implementation of log10.
33*f3087befSAndrew Turner The implementation is similar to that of math/log, except that:
34*f3087befSAndrew Turner - Polynomials are computed for log10(1+r) with r on same intervals as log.
35*f3087befSAndrew Turner - Lookup parameters are scaled (at runtime) to switch from base e to
36*f3087befSAndrew Turner base 10. Many errors above 1.59 ulp are observed across the whole range of
37*f3087befSAndrew Turner doubles. The greatest observed error is 1.61 ulp, at around 0.965:
38*f3087befSAndrew Turner log10(0x1.dc8710333a29bp-1) got -0x1.fee26884905a6p-6
39*f3087befSAndrew Turner want -0x1.fee26884905a8p-6. */
40*f3087befSAndrew Turner double
log10(double x)41*f3087befSAndrew Turner log10 (double x)
42*f3087befSAndrew Turner {
43*f3087befSAndrew Turner /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
44*f3087befSAndrew Turner double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
45*f3087befSAndrew Turner uint64_t ix, iz, tmp;
46*f3087befSAndrew Turner uint32_t top;
47*f3087befSAndrew Turner int k, i;
48*f3087befSAndrew Turner
49*f3087befSAndrew Turner ix = asuint64 (x);
50*f3087befSAndrew Turner top = top16 (x);
51*f3087befSAndrew Turner
52*f3087befSAndrew Turner if (unlikely (ix - LO < HI - LO))
53*f3087befSAndrew Turner {
54*f3087befSAndrew Turner /* Handle close to 1.0 inputs separately. */
55*f3087befSAndrew Turner /* Fix sign of zero with downward rounding when x==1. */
56*f3087befSAndrew Turner if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
57*f3087befSAndrew Turner return 0;
58*f3087befSAndrew Turner r = x - 1.0;
59*f3087befSAndrew Turner r2 = r * r;
60*f3087befSAndrew Turner r3 = r * r2;
61*f3087befSAndrew Turner y = r3
62*f3087befSAndrew Turner * (B[1] + r * B[2] + r2 * B[3]
63*f3087befSAndrew Turner + r3
64*f3087befSAndrew Turner * (B[4] + r * B[5] + r2 * B[6]
65*f3087befSAndrew Turner + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
66*f3087befSAndrew Turner /* Worst-case error is around 0.507 ULP. */
67*f3087befSAndrew Turner w = r * 0x1p27;
68*f3087befSAndrew Turner double_t rhi = r + w - w;
69*f3087befSAndrew Turner double_t rlo = r - rhi;
70*f3087befSAndrew Turner w = rhi * rhi * B[0];
71*f3087befSAndrew Turner hi = r + w;
72*f3087befSAndrew Turner lo = r - hi + w;
73*f3087befSAndrew Turner lo += B[0] * rlo * (rhi + r);
74*f3087befSAndrew Turner y += lo;
75*f3087befSAndrew Turner y += hi;
76*f3087befSAndrew Turner /* Scale by 1/ln(10). Polynomial already contains scaling. */
77*f3087befSAndrew Turner y = y * InvLn10;
78*f3087befSAndrew Turner
79*f3087befSAndrew Turner return eval_as_double (y);
80*f3087befSAndrew Turner }
81*f3087befSAndrew Turner if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
82*f3087befSAndrew Turner {
83*f3087befSAndrew Turner /* x < 0x1p-1022 or inf or nan. */
84*f3087befSAndrew Turner if (ix * 2 == 0)
85*f3087befSAndrew Turner return __math_divzero (1);
86*f3087befSAndrew Turner if (ix == asuint64 (INFINITY)) /* log10(inf) == inf. */
87*f3087befSAndrew Turner return x;
88*f3087befSAndrew Turner if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
89*f3087befSAndrew Turner return __math_invalid (x);
90*f3087befSAndrew Turner /* x is subnormal, normalize it. */
91*f3087befSAndrew Turner ix = asuint64 (x * 0x1p52);
92*f3087befSAndrew Turner ix -= 52ULL << 52;
93*f3087befSAndrew Turner }
94*f3087befSAndrew Turner
95*f3087befSAndrew Turner /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
96*f3087befSAndrew Turner The range is split into N subintervals.
97*f3087befSAndrew Turner The ith subinterval contains z and c is near its center. */
98*f3087befSAndrew Turner tmp = ix - OFF;
99*f3087befSAndrew Turner i = (tmp >> (52 - LOG10_TABLE_BITS)) % N;
100*f3087befSAndrew Turner k = (int64_t) tmp >> 52; /* arithmetic shift. */
101*f3087befSAndrew Turner iz = ix - (tmp & 0xfffULL << 52);
102*f3087befSAndrew Turner invc = T[i].invc;
103*f3087befSAndrew Turner logc = T[i].logc;
104*f3087befSAndrew Turner z = asdouble (iz);
105*f3087befSAndrew Turner
106*f3087befSAndrew Turner /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
107*f3087befSAndrew Turner /* r ~= z/c - 1, |r| < 1/(2*N). */
108*f3087befSAndrew Turner #if HAVE_FAST_FMA
109*f3087befSAndrew Turner /* rounding error: 0x1p-55/N. */
110*f3087befSAndrew Turner r = fma (z, invc, -1.0);
111*f3087befSAndrew Turner #else
112*f3087befSAndrew Turner /* rounding error: 0x1p-55/N + 0x1p-66. */
113*f3087befSAndrew Turner r = (z - T2[i].chi - T2[i].clo) * invc;
114*f3087befSAndrew Turner #endif
115*f3087befSAndrew Turner kd = (double_t) k;
116*f3087befSAndrew Turner
117*f3087befSAndrew Turner /* w = log(c) + k*Ln2hi. */
118*f3087befSAndrew Turner w = kd * Ln2hi + logc;
119*f3087befSAndrew Turner hi = w + r;
120*f3087befSAndrew Turner lo = w - hi + r + kd * Ln2lo;
121*f3087befSAndrew Turner
122*f3087befSAndrew Turner /* log10(x) = (w + r)/log(10) + (log10(1+r) - r/log(10)). */
123*f3087befSAndrew Turner r2 = r * r; /* rounding error: 0x1p-54/N^2. */
124*f3087befSAndrew Turner
125*f3087befSAndrew Turner /* Scale by 1/ln(10). Polynomial already contains scaling. */
126*f3087befSAndrew Turner y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4]))
127*f3087befSAndrew Turner + hi;
128*f3087befSAndrew Turner y = y * InvLn10;
129*f3087befSAndrew Turner
130*f3087befSAndrew Turner return eval_as_double (y);
131*f3087befSAndrew Turner }
132*f3087befSAndrew Turner
133*f3087befSAndrew Turner // clang-format off
134*f3087befSAndrew Turner #if USE_GLIBC_ABI
strong_alias(log10,__log10_finite)135*f3087befSAndrew Turner strong_alias (log10, __log10_finite)
136*f3087befSAndrew Turner hidden_alias (log10, __ieee754_log10)
137*f3087befSAndrew Turner #if LDBL_MANT_DIG == 53
138*f3087befSAndrew Turner long double
139*f3087befSAndrew Turner log10l (long double x)
140*f3087befSAndrew Turner {
141*f3087befSAndrew Turner return log10 (x);
142*f3087befSAndrew Turner }
143*f3087befSAndrew Turner #endif
144*f3087befSAndrew Turner #endif
145*f3087befSAndrew Turner // clang-format on
146*f3087befSAndrew Turner
147*f3087befSAndrew Turner TEST_SIG (S, D, 1, log10, 0.01, 11.1)
148*f3087befSAndrew Turner TEST_ULP (log10, 1.11)
149*f3087befSAndrew Turner TEST_INTERVAL (log10, 0, 0xffff000000000000, 10000)
150*f3087befSAndrew Turner TEST_INTERVAL (log10, 0x1p-4, 0x1p4, 40000)
151*f3087befSAndrew Turner TEST_INTERVAL (log10, 0, inf, 40000)
152