1 /* 2 * Double-precision vector 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 "v_math.h" 9 #include "poly_advsimd_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 float64x2_t poly[12]; 16 float64x2_t pi, pi_over_2; 17 uint64x2_t abs_mask; 18 } data = { 19 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) 20 on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */ 21 .poly = { V2 (0x1.555555555554ep-3), V2 (0x1.3333333337233p-4), 22 V2 (0x1.6db6db67f6d9fp-5), V2 (0x1.f1c71fbd29fbbp-6), 23 V2 (0x1.6e8b264d467d6p-6), V2 (0x1.1c5997c357e9dp-6), 24 V2 (0x1.c86a22cd9389dp-7), V2 (0x1.856073c22ebbep-7), 25 V2 (0x1.fd1151acb6bedp-8), V2 (0x1.087182f799c1dp-6), 26 V2 (-0x1.6602748120927p-7), V2 (0x1.cfa0dd1f9478p-6), }, 27 .pi = V2 (0x1.921fb54442d18p+1), 28 .pi_over_2 = V2 (0x1.921fb54442d18p+0), 29 .abs_mask = V2 (0x7fffffffffffffff), 30 }; 31 32 #define AllMask v_u64 (0xffffffffffffffff) 33 #define Oneu (0x3ff0000000000000) 34 #define Small (0x3e50000000000000) /* 2^-53. */ 35 36 #if WANT_SIMD_EXCEPT 37 static float64x2_t VPCS_ATTR NOINLINE 38 special_case (float64x2_t x, float64x2_t y, uint64x2_t special) 39 { 40 return v_call_f64 (acos, x, y, special); 41 } 42 #endif 43 44 /* Double-precision implementation of vector acos(x). 45 46 For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-53 for correct 47 rounding. 48 If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following 49 approximation. 50 51 For |x| in [Small, 0.5], use an order 11 polynomial P such that the final 52 approximation of asin is an odd polynomial: 53 54 acos(x) ~ pi/2 - (x + x^3 P(x^2)). 55 56 The largest observed error in this region is 1.18 ulps, 57 _ZGVnN2v_acos (0x1.fbab0a7c460f6p-2) got 0x1.0d54d1985c068p+0 58 want 0x1.0d54d1985c069p+0. 59 60 For |x| in [0.5, 1.0], use same approximation with a change of variable 61 62 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 63 64 The largest observed error in this region is 1.52 ulps, 65 _ZGVnN2v_acos (0x1.23d362722f591p-1) got 0x1.edbbedf8a7d6ep-1 66 want 0x1.edbbedf8a7d6cp-1. */ 67 float64x2_t VPCS_ATTR V_NAME_D1 (acos) (float64x2_t x) 68 { 69 const struct data *d = ptr_barrier (&data); 70 71 float64x2_t ax = vabsq_f64 (x); 72 73 #if WANT_SIMD_EXCEPT 74 /* A single comparison for One, Small and QNaN. */ 75 uint64x2_t special 76 = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)), 77 v_u64 (Oneu - Small)); 78 if (unlikely (v_any_u64 (special))) 79 return special_case (x, x, AllMask); 80 #endif 81 82 uint64x2_t a_le_half = vcleq_f64 (ax, v_f64 (0.5)); 83 84 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 85 z2 = x ^ 2 and z = |x| , if |x| < 0.5 86 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 87 float64x2_t z2 = vbslq_f64 (a_le_half, vmulq_f64 (x, x), 88 vfmaq_f64 (v_f64 (0.5), v_f64 (-0.5), ax)); 89 float64x2_t z = vbslq_f64 (a_le_half, ax, vsqrtq_f64 (z2)); 90 91 /* Use a single polynomial approximation P for both intervals. */ 92 float64x2_t z4 = vmulq_f64 (z2, z2); 93 float64x2_t z8 = vmulq_f64 (z4, z4); 94 float64x2_t z16 = vmulq_f64 (z8, z8); 95 float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly); 96 97 /* Finalize polynomial: z + z * z2 * P(z2). */ 98 p = vfmaq_f64 (z, vmulq_f64 (z, z2), p); 99 100 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 101 = 2 Q(|x|) , for 0.5 < x < 1.0 102 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 103 float64x2_t y = vbslq_f64 (d->abs_mask, p, x); 104 105 uint64x2_t is_neg = vcltzq_f64 (x); 106 float64x2_t off = vreinterpretq_f64_u64 ( 107 vandq_u64 (is_neg, vreinterpretq_u64_f64 (d->pi))); 108 float64x2_t mul = vbslq_f64 (a_le_half, v_f64 (-1.0), v_f64 (2.0)); 109 float64x2_t add = vbslq_f64 (a_le_half, d->pi_over_2, off); 110 111 return vfmaq_f64 (add, mul, y); 112 } 113 114 PL_SIG (V, D, 1, acos, -1.0, 1.0) 115 PL_TEST_ULP (V_NAME_D1 (acos), 1.02) 116 PL_TEST_EXPECT_FENV (V_NAME_D1 (acos), WANT_SIMD_EXCEPT) 117 PL_TEST_INTERVAL (V_NAME_D1 (acos), 0, Small, 5000) 118 PL_TEST_INTERVAL (V_NAME_D1 (acos), Small, 0.5, 50000) 119 PL_TEST_INTERVAL (V_NAME_D1 (acos), 0.5, 1.0, 50000) 120 PL_TEST_INTERVAL (V_NAME_D1 (acos), 1.0, 0x1p11, 50000) 121 PL_TEST_INTERVAL (V_NAME_D1 (acos), 0x1p11, inf, 20000) 122 PL_TEST_INTERVAL (V_NAME_D1 (acos), -0, -inf, 20000) 123