1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Single-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_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, pif; 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 .pif = V4 (0x1.921fb6p+1f), 24*f3087befSAndrew Turner }; 25*f3087befSAndrew Turner 26*f3087befSAndrew Turner #define AbsMask 0x7fffffff 27*f3087befSAndrew Turner #define Half 0x3f000000 28*f3087befSAndrew Turner #define One 0x3f800000 29*f3087befSAndrew Turner #define Small 0x32800000 /* 2^-26. */ 30*f3087befSAndrew Turner 31*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 32*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE 33*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint32x4_t special) 34*f3087befSAndrew Turner { 35*f3087befSAndrew Turner return v_call_f32 (acosf, x, y, special); 36*f3087befSAndrew Turner } 37*f3087befSAndrew Turner #endif 38*f3087befSAndrew Turner 39*f3087befSAndrew Turner /* Single-precision implementation of vector acos(x). 40*f3087befSAndrew Turner 41*f3087befSAndrew Turner For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct 42*f3087befSAndrew Turner rounding. 43*f3087befSAndrew Turner If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following 44*f3087befSAndrew Turner approximation. 45*f3087befSAndrew Turner 46*f3087befSAndrew Turner For |x| in [Small, 0.5], use order 4 polynomial P such that the final 47*f3087befSAndrew Turner approximation of asin is an odd polynomial: 48*f3087befSAndrew Turner 49*f3087befSAndrew Turner acos(x) ~ pi/2 - (x + x^3 P(x^2)). 50*f3087befSAndrew Turner 51*f3087befSAndrew Turner The largest observed error in this region is 1.26 ulps, 52*f3087befSAndrew Turner _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0. 53*f3087befSAndrew Turner 54*f3087befSAndrew Turner For |x| in [0.5, 1.0], use same approximation with a change of variable 55*f3087befSAndrew Turner 56*f3087befSAndrew Turner acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 57*f3087befSAndrew Turner 58*f3087befSAndrew Turner The largest observed error in this region is 1.32 ulps, 59*f3087befSAndrew Turner _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1 60*f3087befSAndrew Turner want 0x1.feb32ep-1. */ 61*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (acos) (float32x4_t x) 62*f3087befSAndrew Turner { 63*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 64*f3087befSAndrew Turner 65*f3087befSAndrew Turner uint32x4_t ix = vreinterpretq_u32_f32 (x); 66*f3087befSAndrew Turner uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask)); 67*f3087befSAndrew Turner 68*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 69*f3087befSAndrew Turner /* A single comparison for One, Small and QNaN. */ 70*f3087befSAndrew Turner uint32x4_t special 71*f3087befSAndrew Turner = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small)); 72*f3087befSAndrew Turner if (unlikely (v_any_u32 (special))) 73*f3087befSAndrew Turner return special_case (x, x, v_u32 (0xffffffff)); 74*f3087befSAndrew Turner #endif 75*f3087befSAndrew Turner 76*f3087befSAndrew Turner float32x4_t ax = vreinterpretq_f32_u32 (ia); 77*f3087befSAndrew Turner uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half)); 78*f3087befSAndrew Turner 79*f3087befSAndrew Turner /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 80*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5 81*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 82*f3087befSAndrew Turner float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x), 83*f3087befSAndrew Turner vfmsq_n_f32 (v_f32 (0.5), ax, 0.5)); 84*f3087befSAndrew Turner float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2)); 85*f3087befSAndrew Turner 86*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */ 87*f3087befSAndrew Turner float32x4_t p = v_horner_4_f32 (z2, d->poly); 88*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */ 89*f3087befSAndrew Turner p = vfmaq_f32 (z, vmulq_f32 (z, z2), p); 90*f3087befSAndrew Turner 91*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 92*f3087befSAndrew Turner = 2 Q(|x|) , for 0.5 < x < 1.0 93*f3087befSAndrew Turner = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 94*f3087befSAndrew Turner float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x); 95*f3087befSAndrew Turner 96*f3087befSAndrew Turner uint32x4_t is_neg = vcltzq_f32 (x); 97*f3087befSAndrew Turner float32x4_t off = vreinterpretq_f32_u32 ( 98*f3087befSAndrew Turner vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg)); 99*f3087befSAndrew Turner float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0)); 100*f3087befSAndrew Turner float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off); 101*f3087befSAndrew Turner 102*f3087befSAndrew Turner return vfmaq_f32 (add, mul, y); 103*f3087befSAndrew Turner } 104*f3087befSAndrew Turner 105*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (acos) 106*f3087befSAndrew Turner 107*f3087befSAndrew Turner TEST_SIG (V, F, 1, acos, -1.0, 1.0) 108*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (acos), 0.82) 109*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (acos), WANT_SIMD_EXCEPT) 110*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0, 0x1p-26, 5000) 111*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0x1p-26, 0.5, 50000) 112*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0.5, 1.0, 50000) 113*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 1.0, 0x1p11, 50000) 114*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0x1p11, inf, 20000) 115*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), -0, -inf, 20000) 116