1 /* 2 * Double-precision acos(x) function. 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "math_config.h" 9 #include "poly_scalar_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define AbsMask (0x7fffffffffffffff) 14 #define Half (0x3fe0000000000000) 15 #define One (0x3ff0000000000000) 16 #define PiOver2 (0x1.921fb54442d18p+0) 17 #define Pi (0x1.921fb54442d18p+1) 18 #define Small (0x3c90000000000000) /* 2^-53. */ 19 #define Small16 (0x3c90) 20 #define QNaN (0x7ff8) 21 22 /* Fast implementation of double-precision acos(x) based on polynomial 23 approximation of double-precision asin(x). 24 25 For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct 26 rounding. 27 28 For |x| in [Small, 0.5], use the trigonometric identity 29 30 acos(x) = pi/2 - asin(x) 31 32 and use an order 11 polynomial P such that the final approximation of asin is 33 an odd polynomial: asin(x) ~ x + x^3 * P(x^2). 34 35 The largest observed error in this region is 1.18 ulps, 36 acos(0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0 37 want 0x1.0d54d1985c069p+0. 38 39 For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1 40 41 acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)) 42 43 where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the 44 approximation of asin near 0. 45 46 The largest observed error in this region is 1.52 ulps, 47 acos(0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1 48 want 0x1.edbbedf8a7d6cp-1. 49 50 For x in [-1.0, -0.5], use this other identity to deduce the negative inputs 51 from their absolute value: acos(x) = pi - acos(-x). */ 52 double 53 acos (double x) 54 { 55 uint64_t ix = asuint64 (x); 56 uint64_t ia = ix & AbsMask; 57 uint64_t ia16 = ia >> 48; 58 double ax = asdouble (ia); 59 uint64_t sign = ix & ~AbsMask; 60 61 /* Special values and invalid range. */ 62 if (unlikely (ia16 == QNaN)) 63 return x; 64 if (ia > One) 65 return __math_invalid (x); 66 if (ia16 < Small16) 67 return PiOver2 - x; 68 69 /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with 70 z2 = x ^ 2 and z = |x| , if |x| < 0.5 71 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 72 double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5); 73 double z = ax < 0.5 ? ax : sqrt (z2); 74 75 /* Use a single polynomial approximation P for both intervals. */ 76 double z4 = z2 * z2; 77 double z8 = z4 * z4; 78 double z16 = z8 * z8; 79 double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly); 80 81 /* Finalize polynomial: z + z * z2 * P(z2). */ 82 p = fma (z * z2, p, z); 83 84 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 85 = pi - 2 Q(|x|), for -1.0 < x <= -0.5 86 = 2 Q(|x|) , for -0.5 < x < 0.0. */ 87 if (ax < 0.5) 88 return PiOver2 - asdouble (asuint64 (p) | sign); 89 90 return (x <= -0.5) ? fma (-2.0, p, Pi) : 2.0 * p; 91 } 92 93 PL_SIG (S, D, 1, acos, -1.0, 1.0) 94 PL_TEST_ULP (acos, 1.02) 95 PL_TEST_INTERVAL (acos, 0, Small, 5000) 96 PL_TEST_INTERVAL (acos, Small, 0.5, 50000) 97 PL_TEST_INTERVAL (acos, 0.5, 1.0, 50000) 98 PL_TEST_INTERVAL (acos, 1.0, 0x1p11, 50000) 99 PL_TEST_INTERVAL (acos, 0x1p11, inf, 20000) 100 PL_TEST_INTERVAL (acos, -0, -inf, 20000) 101