1 /* 2 * Single-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 "poly_scalar_f32.h" 9 #include "math_config.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define AbsMask (0x7fffffff) 14 #define Half (0x3f000000) 15 #define One (0x3f800000) 16 #define PiOver2f (0x1.921fb6p+0f) 17 #define Pif (0x1.921fb6p+1f) 18 #define Small (0x32800000) /* 2^-26. */ 19 #define Small12 (0x328) 20 #define QNaN (0x7fc) 21 22 /* Fast implementation of single-precision acos(x) based on polynomial 23 approximation of single-precision asin(x). 24 25 For x < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 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 4 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 1.16 ulps, 36 acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0 want 0x1.0c27f6p+0. 37 38 For |x| in [0.5, 1.0], use the following development of acos(x) near x = 1 39 40 acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)) 41 42 where z = (1-x)/2, z is near 0 when x approaches 1, and P contributes to the 43 approximation of asin near 0. 44 45 The largest observed error in this region is 1.32 ulps, 46 acosf(0x1.15ba56p-1) got 0x1.feb33p-1 want 0x1.feb32ep-1. 47 48 For x in [-1.0, -0.5], use this other identity to deduce the negative inputs 49 from their absolute value. 50 51 acos(x) = pi - acos(-x) 52 53 The largest observed error in this region is 1.28 ulps, 54 acosf(-0x1.002072p-1) got 0x1.0c1e84p+1 want 0x1.0c1e82p+1. */ 55 float 56 acosf (float x) 57 { 58 uint32_t ix = asuint (x); 59 uint32_t ia = ix & AbsMask; 60 uint32_t ia12 = ia >> 20; 61 float ax = asfloat (ia); 62 uint32_t sign = ix & ~AbsMask; 63 64 /* Special values and invalid range. */ 65 if (unlikely (ia12 == QNaN)) 66 return x; 67 if (ia > One) 68 return __math_invalidf (x); 69 if (ia12 < Small12) 70 return PiOver2f - x; 71 72 /* Evaluate polynomial Q(|x|) = z + z * z2 * P(z2) with 73 z2 = x ^ 2 and z = |x| , if |x| < 0.5 74 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 75 float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f); 76 float z = ax < 0.5 ? ax : sqrtf (z2); 77 78 /* Use a single polynomial approximation P for both intervals. */ 79 float p = horner_4_f32 (z2, __asinf_poly); 80 /* Finalize polynomial: z + z * z2 * P(z2). */ 81 p = fmaf (z * z2, p, z); 82 83 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 84 = pi - 2 Q(|x|), for -1.0 < x <= -0.5 85 = 2 Q(|x|) , for -0.5 < x < 0.0. */ 86 if (ax < 0.5) 87 return PiOver2f - asfloat (asuint (p) | sign); 88 89 return (x <= -0.5) ? fmaf (-2.0f, p, Pif) : 2.0f * p; 90 } 91 92 PL_SIG (S, F, 1, acos, -1.0, 1.0) 93 PL_TEST_ULP (acosf, 0.82) 94 PL_TEST_INTERVAL (acosf, 0, Small, 5000) 95 PL_TEST_INTERVAL (acosf, Small, 0.5, 50000) 96 PL_TEST_INTERVAL (acosf, 0.5, 1.0, 50000) 97 PL_TEST_INTERVAL (acosf, 1.0, 0x1p11, 50000) 98 PL_TEST_INTERVAL (acosf, 0x1p11, inf, 20000) 99 PL_TEST_INTERVAL (acosf, -0, -inf, 20000) 100