1 /* 2 * Double-precision SVE asin(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_over_2f; 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, 21 0x1.6db6db67f6d9fp-5, 0x1.f1c71fbd29fbbp-6, 22 0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6, 23 0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7, 24 0x1.fd1151acb6bedp-8, 0x1.087182f799c1dp-6, 25 -0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, }, 26 .pi_over_2f = 0x1.921fb54442d18p+0, 27 }; 28 29 #define P(i) sv_f64 (d->poly[i]) 30 31 /* Double-precision SVE implementation of vector asin(x). 32 33 For |x| in [0, 0.5], use an order 11 polynomial P such that the final 34 approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). 35 36 The largest observed error in this region is 0.52 ulps, 37 _ZGVsMxv_asin(0x1.d95ae04998b6cp-2) got 0x1.ec13757305f27p-2 38 want 0x1.ec13757305f26p-2. 39 40 For |x| in [0.5, 1.0], use same approximation with a change of variable 41 42 asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z). 43 44 The largest observed error in this region is 2.69 ulps, 45 _ZGVsMxv_asin(0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1 46 want 0x1.110d7e85fdd53p-1. */ 47 svfloat64_t SV_NAME_D1 (asin) (svfloat64_t x, const svbool_t pg) 48 { 49 const struct data *d = ptr_barrier (&data); 50 51 svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000); 52 svfloat64_t ax = svabs_x (pg, x); 53 svbool_t a_ge_half = svacge (pg, x, 0.5); 54 55 /* Evaluate polynomial Q(x) = y + y * z * P(z) with 56 z = x ^ 2 and y = |x| , if |x| < 0.5 57 z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */ 58 svfloat64_t z2 = svsel (a_ge_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_ge_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 /* Finalize polynomial: z + z * z2 * P(z2). */ 68 p = svmla_x (pg, z, svmul_x (pg, z, z2), p); 69 70 /* asin(|x|) = Q(|x|) , for |x| < 0.5 71 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ 72 svfloat64_t y = svmad_m (a_ge_half, p, sv_f64 (-2.0), d->pi_over_2f); 73 74 /* Copy sign. */ 75 return svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign)); 76 } 77 78 PL_SIG (SV, D, 1, asin, -1.0, 1.0) 79 PL_TEST_ULP (SV_NAME_D1 (asin), 2.19) 80 PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0, 0.5, 50000) 81 PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0.5, 1.0, 50000) 82 PL_TEST_INTERVAL (SV_NAME_D1 (asin), 1.0, 0x1p11, 50000) 83 PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0x1p11, inf, 20000) 84 PL_TEST_INTERVAL (SV_NAME_D1 (asin), -0, -inf, 20000) 85