1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision acos(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2023-2024, 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 "poly_scalar_f64.h" 10*f3087befSAndrew Turner #include "test_sig.h" 11*f3087befSAndrew Turner #include "test_defs.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff 14*f3087befSAndrew Turner #define Half 0x3fe0000000000000 15*f3087befSAndrew Turner #define One 0x3ff0000000000000 16*f3087befSAndrew Turner #define PiOver2 0x1.921fb54442d18p+0 17*f3087befSAndrew Turner #define Pi 0x1.921fb54442d18p+1 18*f3087befSAndrew Turner #define Small 0x3c90000000000000 /* 2^-53. */ 19*f3087befSAndrew Turner #define Small16 0x3c90 20*f3087befSAndrew Turner #define QNaN 0x7ff8 21*f3087befSAndrew Turner 22*f3087befSAndrew Turner /* Fast implementation of double-precision acos(x) based on polynomial 23*f3087befSAndrew Turner approximation of double-precision asin(x). 24*f3087befSAndrew Turner 25*f3087befSAndrew Turner For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct 26*f3087befSAndrew Turner rounding. 27*f3087befSAndrew Turner 28*f3087befSAndrew Turner For |x| in [Small, 0.5], use the trigonometric identity 29*f3087befSAndrew Turner 30*f3087befSAndrew Turner acos(x) = pi/2 - asin(x) 31*f3087befSAndrew Turner 32*f3087befSAndrew Turner and use an order 11 polynomial P such that the final approximation of asin 33*f3087befSAndrew Turner is an odd polynomial: asin(x) ~ x + x^3 * P(x^2). 34*f3087befSAndrew Turner 35*f3087befSAndrew Turner The largest observed error in this region is 1.18 ulps, 36*f3087befSAndrew Turner acos(0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0 37*f3087befSAndrew Turner want 0x1.0d54d1985c069p+0. 38*f3087befSAndrew Turner 39*f3087befSAndrew Turner For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1 40*f3087befSAndrew Turner 41*f3087befSAndrew Turner acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)) 42*f3087befSAndrew Turner 43*f3087befSAndrew Turner where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the 44*f3087befSAndrew Turner approximation of asin near 0. 45*f3087befSAndrew Turner 46*f3087befSAndrew Turner The largest observed error in this region is 1.52 ulps, 47*f3087befSAndrew Turner acos(0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1 48*f3087befSAndrew Turner want 0x1.edbbedf8a7d6cp-1. 49*f3087befSAndrew Turner 50*f3087befSAndrew Turner For x in [-1.0, -0.5], use this other identity to deduce the negative inputs 51*f3087befSAndrew Turner from their absolute value: acos(x) = pi - acos(-x). */ 52*f3087befSAndrew Turner double 53*f3087befSAndrew Turner acos (double x) 54*f3087befSAndrew Turner { 55*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 56*f3087befSAndrew Turner uint64_t ia = ix & AbsMask; 57*f3087befSAndrew Turner uint64_t ia16 = ia >> 48; 58*f3087befSAndrew Turner double ax = asdouble (ia); 59*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask; 60*f3087befSAndrew Turner 61*f3087befSAndrew Turner /* Special values and invalid range. */ 62*f3087befSAndrew Turner if (unlikely (ia16 == QNaN)) 63*f3087befSAndrew Turner return x; 64*f3087befSAndrew Turner if (ia > One) 65*f3087befSAndrew Turner return __math_invalid (x); 66*f3087befSAndrew Turner if (ia16 < Small16) 67*f3087befSAndrew Turner return PiOver2 - x; 68*f3087befSAndrew Turner 69*f3087befSAndrew Turner /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with 70*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5 71*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 72*f3087befSAndrew Turner double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5); 73*f3087befSAndrew Turner double z = ax < 0.5 ? ax : sqrt (z2); 74*f3087befSAndrew Turner 75*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */ 76*f3087befSAndrew Turner double z4 = z2 * z2; 77*f3087befSAndrew Turner double z8 = z4 * z4; 78*f3087befSAndrew Turner double z16 = z8 * z8; 79*f3087befSAndrew Turner double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly); 80*f3087befSAndrew Turner 81*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */ 82*f3087befSAndrew Turner p = fma (z * z2, p, z); 83*f3087befSAndrew Turner 84*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 85*f3087befSAndrew Turner = pi - 2 Q(|x|), for -1.0 < x <= -0.5 86*f3087befSAndrew Turner = 2 Q(|x|) , for -0.5 < x < 0.0. */ 87*f3087befSAndrew Turner if (ax < 0.5) 88*f3087befSAndrew Turner return PiOver2 - asdouble (asuint64 (p) | sign); 89*f3087befSAndrew Turner 90*f3087befSAndrew Turner return (x <= -0.5) ? fma (-2.0, p, Pi) : 2.0 * p; 91*f3087befSAndrew Turner } 92*f3087befSAndrew Turner 93*f3087befSAndrew Turner TEST_SIG (S, D, 1, acos, -1.0, 1.0) 94*f3087befSAndrew Turner TEST_ULP (acos, 1.02) 95*f3087befSAndrew Turner TEST_INTERVAL (acos, 0, Small, 5000) 96*f3087befSAndrew Turner TEST_INTERVAL (acos, Small, 0.5, 50000) 97*f3087befSAndrew Turner TEST_INTERVAL (acos, 0.5, 1.0, 50000) 98*f3087befSAndrew Turner TEST_INTERVAL (acos, 1.0, 0x1p11, 50000) 99*f3087befSAndrew Turner TEST_INTERVAL (acos, 0x1p11, inf, 20000) 100*f3087befSAndrew Turner TEST_INTERVAL (acos, -0, -inf, 20000) 101