1 /* 2 * Double-precision SVE 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 "sv_math.h" 9 #include "poly_sve_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 float64_t poly[12]; 16 float64_t pi, pi_over_2; 17 } data = { 18 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) 19 on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */ 20 .poly = { 0x1.555555555554ep-3, 0x1.3333333337233p-4, 0x1.6db6db67f6d9fp-5, 21 0x1.f1c71fbd29fbbp-6, 0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6, 22 0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7, 0x1.fd1151acb6bedp-8, 23 0x1.087182f799c1dp-6, -0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, }, 24 .pi = 0x1.921fb54442d18p+1, 25 .pi_over_2 = 0x1.921fb54442d18p+0, 26 }; 27 28 /* Double-precision SVE implementation of vector acos(x). 29 30 For |x| in [0, 0.5], use an order 11 polynomial P such that the final 31 approximation of asin is an odd polynomial: 32 33 acos(x) ~ pi/2 - (x + x^3 P(x^2)). 34 35 The largest observed error in this region is 1.18 ulps, 36 _ZGVsMxv_acos (0x1.fbc5fe28ee9e3p-2) got 0x1.0d4d0f55667f6p+0 37 want 0x1.0d4d0f55667f7p+0. 38 39 For |x| in [0.5, 1.0], use same approximation with a change of variable 40 41 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 42 43 The largest observed error in this region is 1.52 ulps, 44 _ZGVsMxv_acos (0x1.24024271a500ap-1) got 0x1.ed82df4243f0dp-1 45 want 0x1.ed82df4243f0bp-1. */ 46 svfloat64_t SV_NAME_D1 (acos) (svfloat64_t x, const svbool_t pg) 47 { 48 const struct data *d = ptr_barrier (&data); 49 50 svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000); 51 svfloat64_t ax = svabs_x (pg, x); 52 53 svbool_t a_gt_half = svacgt (pg, x, 0.5); 54 55 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 56 z2 = x ^ 2 and z = |x| , if |x| < 0.5 57 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 58 svfloat64_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f64 (0.5), ax, 0.5), 59 svmul_x (pg, x, x)); 60 svfloat64_t z = svsqrt_m (ax, a_gt_half, z2); 61 62 /* Use a single polynomial approximation P for both intervals. */ 63 svfloat64_t z4 = svmul_x (pg, z2, z2); 64 svfloat64_t z8 = svmul_x (pg, z4, z4); 65 svfloat64_t z16 = svmul_x (pg, z8, z8); 66 svfloat64_t p = sv_estrin_11_f64_x (pg, z2, z4, z8, z16, d->poly); 67 68 /* Finalize polynomial: z + z * z2 * P(z2). */ 69 p = svmla_x (pg, z, svmul_x (pg, z, z2), p); 70 71 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 72 = 2 Q(|x|) , for 0.5 < x < 1.0 73 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 74 svfloat64_t y 75 = svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (p), sign)); 76 77 svbool_t is_neg = svcmplt (pg, x, 0.0); 78 svfloat64_t off = svdup_f64_z (is_neg, d->pi); 79 svfloat64_t mul = svsel (a_gt_half, sv_f64 (2.0), sv_f64 (-1.0)); 80 svfloat64_t add = svsel (a_gt_half, off, sv_f64 (d->pi_over_2)); 81 82 return svmla_x (pg, add, mul, y); 83 } 84 85 PL_SIG (SV, D, 1, acos, -1.0, 1.0) 86 PL_TEST_ULP (SV_NAME_D1 (acos), 1.02) 87 PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0, 0.5, 50000) 88 PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0.5, 1.0, 50000) 89 PL_TEST_INTERVAL (SV_NAME_D1 (acos), 1.0, 0x1p11, 50000) 90 PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0x1p11, inf, 20000) 91 PL_TEST_INTERVAL (SV_NAME_D1 (acos), -0, -inf, 20000) 92