1 /* 2 * Double-precision vector 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 "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_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_over_2 = V2 (0x1.921fb54442d18p+0), 28 .abs_mask = V2 (0x7fffffffffffffff), 29 }; 30 31 #define AllMask v_u64 (0xffffffffffffffff) 32 #define One (0x3ff0000000000000) 33 #define Small (0x3e50000000000000) /* 2^-12. */ 34 35 #if WANT_SIMD_EXCEPT 36 static float64x2_t VPCS_ATTR NOINLINE 37 special_case (float64x2_t x, float64x2_t y, uint64x2_t special) 38 { 39 return v_call_f64 (asin, x, y, special); 40 } 41 #endif 42 43 /* Double-precision implementation of vector asin(x). 44 45 For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct 46 rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the 47 following approximation. 48 49 For |x| in [Small, 0.5], use an order 11 polynomial P such that the final 50 approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). 51 52 The largest observed error in this region is 1.01 ulps, 53 _ZGVnN2v_asin (0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2 54 want 0x1.ed78525a927eep-2. 55 56 For |x| in [0.5, 1.0], use same approximation with a change of variable 57 58 asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z). 59 60 The largest observed error in this region is 2.69 ulps, 61 _ZGVnN2v_asin (0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1 62 want 0x1.110d7e85fdd53p-1. */ 63 float64x2_t VPCS_ATTR V_NAME_D1 (asin) (float64x2_t x) 64 { 65 const struct data *d = ptr_barrier (&data); 66 67 float64x2_t ax = vabsq_f64 (x); 68 69 #if WANT_SIMD_EXCEPT 70 /* Special values need to be computed with scalar fallbacks so 71 that appropriate exceptions are raised. */ 72 uint64x2_t special 73 = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)), 74 v_u64 (One - Small)); 75 if (unlikely (v_any_u64 (special))) 76 return special_case (x, x, AllMask); 77 #endif 78 79 uint64x2_t a_lt_half = vcltq_f64 (ax, v_f64 (0.5)); 80 81 /* Evaluate polynomial Q(x) = y + y * z * P(z) with 82 z = x ^ 2 and y = |x| , if |x| < 0.5 83 z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */ 84 float64x2_t z2 = vbslq_f64 (a_lt_half, vmulq_f64 (x, x), 85 vfmsq_n_f64 (v_f64 (0.5), ax, 0.5)); 86 float64x2_t z = vbslq_f64 (a_lt_half, ax, vsqrtq_f64 (z2)); 87 88 /* Use a single polynomial approximation P for both intervals. */ 89 float64x2_t z4 = vmulq_f64 (z2, z2); 90 float64x2_t z8 = vmulq_f64 (z4, z4); 91 float64x2_t z16 = vmulq_f64 (z8, z8); 92 float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly); 93 94 /* Finalize polynomial: z + z * z2 * P(z2). */ 95 p = vfmaq_f64 (z, vmulq_f64 (z, z2), p); 96 97 /* asin(|x|) = Q(|x|) , for |x| < 0.5 98 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ 99 float64x2_t y = vbslq_f64 (a_lt_half, p, vfmsq_n_f64 (d->pi_over_2, p, 2.0)); 100 101 /* Copy sign. */ 102 return vbslq_f64 (d->abs_mask, y, x); 103 } 104 105 PL_SIG (V, D, 1, asin, -1.0, 1.0) 106 PL_TEST_ULP (V_NAME_D1 (asin), 2.19) 107 PL_TEST_EXPECT_FENV (V_NAME_D1 (asin), WANT_SIMD_EXCEPT) 108 PL_TEST_INTERVAL (V_NAME_D1 (asin), 0, Small, 5000) 109 PL_TEST_INTERVAL (V_NAME_D1 (asin), Small, 0.5, 50000) 110 PL_TEST_INTERVAL (V_NAME_D1 (asin), 0.5, 1.0, 50000) 111 PL_TEST_INTERVAL (V_NAME_D1 (asin), 1.0, 0x1p11, 50000) 112 PL_TEST_INTERVAL (V_NAME_D1 (asin), 0x1p11, inf, 20000) 113 PL_TEST_INTERVAL (V_NAME_D1 (asin), -0, -inf, 20000) 114