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 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 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 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