1 /* 2 * Single-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_f32.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 float32_t poly[5]; 16 float32_t pi, pi_over_2; 17 } data = { 18 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on 19 [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */ 20 .poly = { 0x1.55555ep-3, 0x1.33261ap-4, 0x1.70d7dcp-5, 0x1.b059dp-6, 21 0x1.3af7d8p-5, }, 22 .pi = 0x1.921fb6p+1f, 23 .pi_over_2 = 0x1.921fb6p+0f, 24 }; 25 26 /* Single-precision SVE implementation of vector acos(x). 27 28 For |x| in [0, 0.5], use order 4 polynomial P such that the final 29 approximation of asin is an odd polynomial: 30 31 acos(x) ~ pi/2 - (x + x^3 P(x^2)). 32 33 The largest observed error in this region is 1.16 ulps, 34 _ZGVsMxv_acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0 35 want 0x1.0c27f6p+0. 36 37 For |x| in [0.5, 1.0], use same approximation with a change of variable 38 39 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 40 41 The largest observed error in this region is 1.32 ulps, 42 _ZGVsMxv_acosf (0x1.15ba56p-1) got 0x1.feb33p-1 43 want 0x1.feb32ep-1. */ 44 svfloat32_t SV_NAME_F1 (acos) (svfloat32_t x, const svbool_t pg) 45 { 46 const struct data *d = ptr_barrier (&data); 47 48 svuint32_t sign = svand_x (pg, svreinterpret_u32 (x), 0x80000000); 49 svfloat32_t ax = svabs_x (pg, x); 50 svbool_t a_gt_half = svacgt (pg, x, 0.5); 51 52 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 53 z2 = x ^ 2 and z = |x| , if |x| < 0.5 54 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 55 svfloat32_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f32 (0.5), ax, 0.5), 56 svmul_x (pg, x, x)); 57 svfloat32_t z = svsqrt_m (ax, a_gt_half, z2); 58 59 /* Use a single polynomial approximation P for both intervals. */ 60 svfloat32_t p = sv_horner_4_f32_x (pg, z2, d->poly); 61 /* Finalize polynomial: z + z * z2 * P(z2). */ 62 p = svmla_x (pg, z, svmul_x (pg, z, z2), p); 63 64 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 65 = 2 Q(|x|) , for 0.5 < x < 1.0 66 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 67 svfloat32_t y 68 = svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (p), sign)); 69 70 svbool_t is_neg = svcmplt (pg, x, 0.0); 71 svfloat32_t off = svdup_f32_z (is_neg, d->pi); 72 svfloat32_t mul = svsel (a_gt_half, sv_f32 (2.0), sv_f32 (-1.0)); 73 svfloat32_t add = svsel (a_gt_half, off, sv_f32 (d->pi_over_2)); 74 75 return svmla_x (pg, add, mul, y); 76 } 77 78 PL_SIG (SV, F, 1, acos, -1.0, 1.0) 79 PL_TEST_ULP (SV_NAME_F1 (acos), 0.82) 80 PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0, 0.5, 50000) 81 PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0.5, 1.0, 50000) 82 PL_TEST_INTERVAL (SV_NAME_F1 (acos), 1.0, 0x1p11, 50000) 83 PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0x1p11, inf, 20000) 84 PL_TEST_INTERVAL (SV_NAME_F1 (acos), -0, -inf, 20000) 85