1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Single-precision vector asin(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_f32.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 float32x4_t poly[5]; 16*f3087befSAndrew Turner float32x4_t pi_over_2f; 17*f3087befSAndrew Turner } data = { 18*f3087befSAndrew Turner /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on 19*f3087befSAndrew Turner [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */ 20*f3087befSAndrew Turner .poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5), 21*f3087befSAndrew Turner V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) }, 22*f3087befSAndrew Turner .pi_over_2f = V4 (0x1.921fb6p+0f), 23*f3087befSAndrew Turner }; 24*f3087befSAndrew Turner 25*f3087befSAndrew Turner #define AbsMask 0x7fffffff 26*f3087befSAndrew Turner #define Half 0x3f000000 27*f3087befSAndrew Turner #define One 0x3f800000 28*f3087befSAndrew Turner #define Small 0x39800000 /* 2^-12. */ 29*f3087befSAndrew Turner 30*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 31*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE 32*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint32x4_t special) 33*f3087befSAndrew Turner { 34*f3087befSAndrew Turner return v_call_f32 (asinf, x, y, special); 35*f3087befSAndrew Turner } 36*f3087befSAndrew Turner #endif 37*f3087befSAndrew Turner 38*f3087befSAndrew Turner /* Single-precision implementation of vector asin(x). 39*f3087befSAndrew Turner 40*f3087befSAndrew Turner For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct 41*f3087befSAndrew Turner rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the 42*f3087befSAndrew Turner following approximation. 43*f3087befSAndrew Turner 44*f3087befSAndrew Turner For |x| in [Small, 0.5], use order 4 polynomial P such that the final 45*f3087befSAndrew Turner approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). 46*f3087befSAndrew Turner 47*f3087befSAndrew Turner The largest observed error in this region is 0.83 ulps, 48*f3087befSAndrew Turner _ZGVnN4v_asinf (0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2. 49*f3087befSAndrew Turner 50*f3087befSAndrew Turner For |x| in [0.5, 1.0], use same approximation with a change of variable 51*f3087befSAndrew Turner 52*f3087befSAndrew Turner asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z). 53*f3087befSAndrew Turner 54*f3087befSAndrew Turner The largest observed error in this region is 2.41 ulps, 55*f3087befSAndrew Turner _ZGVnN4v_asinf (0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */ 56*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (asin) (float32x4_t x) 57*f3087befSAndrew Turner { 58*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner uint32x4_t ix = vreinterpretq_u32_f32 (x); 61*f3087befSAndrew Turner uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask)); 62*f3087befSAndrew Turner 63*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 64*f3087befSAndrew Turner /* Special values need to be computed with scalar fallbacks so 65*f3087befSAndrew Turner that appropriate fp exceptions are raised. */ 66*f3087befSAndrew Turner uint32x4_t special 67*f3087befSAndrew Turner = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small)); 68*f3087befSAndrew Turner if (unlikely (v_any_u32 (special))) 69*f3087befSAndrew Turner return special_case (x, x, v_u32 (0xffffffff)); 70*f3087befSAndrew Turner #endif 71*f3087befSAndrew Turner 72*f3087befSAndrew Turner float32x4_t ax = vreinterpretq_f32_u32 (ia); 73*f3087befSAndrew Turner uint32x4_t a_lt_half = vcltq_u32 (ia, v_u32 (Half)); 74*f3087befSAndrew Turner 75*f3087befSAndrew Turner /* Evaluate polynomial Q(x) = y + y * z * P(z) with 76*f3087befSAndrew Turner z = x ^ 2 and y = |x| , if |x| < 0.5 77*f3087befSAndrew Turner z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */ 78*f3087befSAndrew Turner float32x4_t z2 = vbslq_f32 (a_lt_half, vmulq_f32 (x, x), 79*f3087befSAndrew Turner vfmsq_n_f32 (v_f32 (0.5), ax, 0.5)); 80*f3087befSAndrew Turner float32x4_t z = vbslq_f32 (a_lt_half, ax, vsqrtq_f32 (z2)); 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */ 83*f3087befSAndrew Turner float32x4_t p = v_horner_4_f32 (z2, d->poly); 84*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */ 85*f3087befSAndrew Turner p = vfmaq_f32 (z, vmulq_f32 (z, z2), p); 86*f3087befSAndrew Turner 87*f3087befSAndrew Turner /* asin(|x|) = Q(|x|) , for |x| < 0.5 88*f3087befSAndrew Turner = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ 89*f3087befSAndrew Turner float32x4_t y 90*f3087befSAndrew Turner = vbslq_f32 (a_lt_half, p, vfmsq_n_f32 (d->pi_over_2f, p, 2.0)); 91*f3087befSAndrew Turner 92*f3087befSAndrew Turner /* Copy sign. */ 93*f3087befSAndrew Turner return vbslq_f32 (v_u32 (AbsMask), y, x); 94*f3087befSAndrew Turner } 95*f3087befSAndrew Turner 96*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (asin) 97*f3087befSAndrew Turner 98*f3087befSAndrew Turner TEST_SIG (V, F, 1, asin, -1.0, 1.0) 99*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (asin), 1.91) 100*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (asin), WANT_SIMD_EXCEPT) 101*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), 0, 0x1p-12, 5000) 102*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), 0x1p-12, 0.5, 50000) 103*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), 0.5, 1.0, 50000) 104*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), 1.0, 0x1p11, 50000) 105*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), 0x1p11, inf, 20000) 106*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (asin), -0, -inf, 20000) 107