1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision vector acos(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "v_math.h" 9*f3087befSAndrew Turner #include "v_poly_f64.h" 10*f3087befSAndrew Turner #include "test_sig.h" 11*f3087befSAndrew Turner #include "test_defs.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner static const struct data 14*f3087befSAndrew Turner { 15*f3087befSAndrew Turner float64x2_t poly[12]; 16*f3087befSAndrew Turner float64x2_t pi, pi_over_2; 17*f3087befSAndrew Turner uint64x2_t abs_mask; 18*f3087befSAndrew Turner } data = { 19*f3087befSAndrew Turner /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) 20*f3087befSAndrew Turner on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */ 21*f3087befSAndrew Turner .poly = { V2 (0x1.555555555554ep-3), V2 (0x1.3333333337233p-4), 22*f3087befSAndrew Turner V2 (0x1.6db6db67f6d9fp-5), V2 (0x1.f1c71fbd29fbbp-6), 23*f3087befSAndrew Turner V2 (0x1.6e8b264d467d6p-6), V2 (0x1.1c5997c357e9dp-6), 24*f3087befSAndrew Turner V2 (0x1.c86a22cd9389dp-7), V2 (0x1.856073c22ebbep-7), 25*f3087befSAndrew Turner V2 (0x1.fd1151acb6bedp-8), V2 (0x1.087182f799c1dp-6), 26*f3087befSAndrew Turner V2 (-0x1.6602748120927p-7), V2 (0x1.cfa0dd1f9478p-6), }, 27*f3087befSAndrew Turner .pi = V2 (0x1.921fb54442d18p+1), 28*f3087befSAndrew Turner .pi_over_2 = V2 (0x1.921fb54442d18p+0), 29*f3087befSAndrew Turner .abs_mask = V2 (0x7fffffffffffffff), 30*f3087befSAndrew Turner }; 31*f3087befSAndrew Turner 32*f3087befSAndrew Turner #define AllMask v_u64 (0xffffffffffffffff) 33*f3087befSAndrew Turner #define Oneu 0x3ff0000000000000 34*f3087befSAndrew Turner #define Small 0x3e50000000000000 /* 2^-53. */ 35*f3087befSAndrew Turner 36*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 37*f3087befSAndrew Turner static float64x2_t VPCS_ATTR NOINLINE 38*f3087befSAndrew Turner special_case (float64x2_t x, float64x2_t y, uint64x2_t special) 39*f3087befSAndrew Turner { 40*f3087befSAndrew Turner return v_call_f64 (acos, x, y, special); 41*f3087befSAndrew Turner } 42*f3087befSAndrew Turner #endif 43*f3087befSAndrew Turner 44*f3087befSAndrew Turner /* Double-precision implementation of vector acos(x). 45*f3087befSAndrew Turner 46*f3087befSAndrew Turner For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct 47*f3087befSAndrew Turner rounding. 48*f3087befSAndrew Turner If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following 49*f3087befSAndrew Turner approximation. 50*f3087befSAndrew Turner 51*f3087befSAndrew Turner For |x| in [Small, 0.5], use an order 11 polynomial P such that the final 52*f3087befSAndrew Turner approximation of asin is an odd polynomial: 53*f3087befSAndrew Turner 54*f3087befSAndrew Turner acos(x) ~ pi/2 - (x + x^3 P(x^2)). 55*f3087befSAndrew Turner 56*f3087befSAndrew Turner The largest observed error in this region is 1.18 ulps, 57*f3087befSAndrew Turner _ZGVnN2v_acos (0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0 58*f3087befSAndrew Turner want 0x1.0d54d1985c069p+0. 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner For |x| in [0.5, 1.0], use same approximation with a change of variable 61*f3087befSAndrew Turner 62*f3087befSAndrew Turner acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 63*f3087befSAndrew Turner 64*f3087befSAndrew Turner The largest observed error in this region is 1.52 ulps, 65*f3087befSAndrew Turner _ZGVnN2v_acos (0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1 66*f3087befSAndrew Turner want 0x1.edbbedf8a7d6cp-1. */ 67*f3087befSAndrew Turner float64x2_t VPCS_ATTR V_NAME_D1 (acos) (float64x2_t x) 68*f3087befSAndrew Turner { 69*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 70*f3087befSAndrew Turner 71*f3087befSAndrew Turner float64x2_t ax = vabsq_f64 (x); 72*f3087befSAndrew Turner 73*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 74*f3087befSAndrew Turner /* A single comparison for One, Small and QNaN. */ 75*f3087befSAndrew Turner uint64x2_t special 76*f3087befSAndrew Turner = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)), 77*f3087befSAndrew Turner v_u64 (Oneu - Small)); 78*f3087befSAndrew Turner if (unlikely (v_any_u64 (special))) 79*f3087befSAndrew Turner return special_case (x, x, AllMask); 80*f3087befSAndrew Turner #endif 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner uint64x2_t a_le_half = vcleq_f64 (ax, v_f64 (0.5)); 83*f3087befSAndrew Turner 84*f3087befSAndrew Turner /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 85*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5 86*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 87*f3087befSAndrew Turner float64x2_t z2 = vbslq_f64 (a_le_half, vmulq_f64 (x, x), 88*f3087befSAndrew Turner vfmaq_f64 (v_f64 (0.5), v_f64 (-0.5), ax)); 89*f3087befSAndrew Turner float64x2_t z = vbslq_f64 (a_le_half, ax, vsqrtq_f64 (z2)); 90*f3087befSAndrew Turner 91*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */ 92*f3087befSAndrew Turner float64x2_t z4 = vmulq_f64 (z2, z2); 93*f3087befSAndrew Turner float64x2_t z8 = vmulq_f64 (z4, z4); 94*f3087befSAndrew Turner float64x2_t z16 = vmulq_f64 (z8, z8); 95*f3087befSAndrew Turner float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly); 96*f3087befSAndrew Turner 97*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */ 98*f3087befSAndrew Turner p = vfmaq_f64 (z, vmulq_f64 (z, z2), p); 99*f3087befSAndrew Turner 100*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 101*f3087befSAndrew Turner = 2 Q(|x|) , for 0.5 < x < 1.0 102*f3087befSAndrew Turner = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 103*f3087befSAndrew Turner float64x2_t y = vbslq_f64 (d->abs_mask, p, x); 104*f3087befSAndrew Turner 105*f3087befSAndrew Turner uint64x2_t is_neg = vcltzq_f64 (x); 106*f3087befSAndrew Turner float64x2_t off = vreinterpretq_f64_u64 ( 107*f3087befSAndrew Turner vandq_u64 (is_neg, vreinterpretq_u64_f64 (d->pi))); 108*f3087befSAndrew Turner float64x2_t mul = vbslq_f64 (a_le_half, v_f64 (-1.0), v_f64 (2.0)); 109*f3087befSAndrew Turner float64x2_t add = vbslq_f64 (a_le_half, d->pi_over_2, off); 110*f3087befSAndrew Turner 111*f3087befSAndrew Turner return vfmaq_f64 (add, mul, y); 112*f3087befSAndrew Turner } 113*f3087befSAndrew Turner 114*f3087befSAndrew Turner TEST_SIG (V, D, 1, acos, -1.0, 1.0) 115*f3087befSAndrew Turner TEST_ULP (V_NAME_D1 (acos), 1.02) 116*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_D1 (acos), WANT_SIMD_EXCEPT) 117*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), 0, Small, 5000) 118*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), Small, 0.5, 50000) 119*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), 0.5, 1.0, 50000) 120*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), 1.0, 0x1p11, 50000) 121*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), 0x1p11, inf, 20000) 122*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (acos), -0, -inf, 20000) 123