1 /* 2 * Single-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_f32.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 float32x4_t poly[5]; 16 float32x4_t pi_over_2f, pif; 17 } data = { 18 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on 19 [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */ 20 .poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5), 21 V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) }, 22 .pi_over_2f = V4 (0x1.921fb6p+0f), 23 .pif = V4 (0x1.921fb6p+1f), 24 }; 25 26 #define AbsMask 0x7fffffff 27 #define Half 0x3f000000 28 #define One 0x3f800000 29 #define Small 0x32800000 /* 2^-26. */ 30 31 #if WANT_SIMD_EXCEPT 32 static float32x4_t VPCS_ATTR NOINLINE 33 special_case (float32x4_t x, float32x4_t y, uint32x4_t special) 34 { 35 return v_call_f32 (acosf, x, y, special); 36 } 37 #endif 38 39 /* Single-precision implementation of vector acos(x). 40 41 For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct 42 rounding. 43 If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following 44 approximation. 45 46 For |x| in [Small, 0.5], use order 4 polynomial P such that the final 47 approximation of asin is an odd polynomial: 48 49 acos(x) ~ pi/2 - (x + x^3 P(x^2)). 50 51 The largest observed error in this region is 1.26 ulps, 52 _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0. 53 54 For |x| in [0.5, 1.0], use same approximation with a change of variable 55 56 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z). 57 58 The largest observed error in this region is 1.32 ulps, 59 _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1 60 want 0x1.feb32ep-1. */ 61 float32x4_t VPCS_ATTR V_NAME_F1 (acos) (float32x4_t x) 62 { 63 const struct data *d = ptr_barrier (&data); 64 65 uint32x4_t ix = vreinterpretq_u32_f32 (x); 66 uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask)); 67 68 #if WANT_SIMD_EXCEPT 69 /* A single comparison for One, Small and QNaN. */ 70 uint32x4_t special 71 = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small)); 72 if (unlikely (v_any_u32 (special))) 73 return special_case (x, x, v_u32 (0xffffffff)); 74 #endif 75 76 float32x4_t ax = vreinterpretq_f32_u32 (ia); 77 uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half)); 78 79 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with 80 z2 = x ^ 2 and z = |x| , if |x| < 0.5 81 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 82 float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x), 83 vfmsq_n_f32 (v_f32 (0.5), ax, 0.5)); 84 float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2)); 85 86 /* Use a single polynomial approximation P for both intervals. */ 87 float32x4_t p = v_horner_4_f32 (z2, d->poly); 88 /* Finalize polynomial: z + z * z2 * P(z2). */ 89 p = vfmaq_f32 (z, vmulq_f32 (z, z2), p); 90 91 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5 92 = 2 Q(|x|) , for 0.5 < x < 1.0 93 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */ 94 float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x); 95 96 uint32x4_t is_neg = vcltzq_f32 (x); 97 float32x4_t off = vreinterpretq_f32_u32 ( 98 vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg)); 99 float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0)); 100 float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off); 101 102 return vfmaq_f32 (add, mul, y); 103 } 104 105 PL_SIG (V, F, 1, acos, -1.0, 1.0) 106 PL_TEST_ULP (V_NAME_F1 (acos), 0.82) 107 PL_TEST_EXPECT_FENV (V_NAME_F1 (acos), WANT_SIMD_EXCEPT) 108 PL_TEST_INTERVAL (V_NAME_F1 (acos), 0, 0x1p-26, 5000) 109 PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p-26, 0.5, 50000) 110 PL_TEST_INTERVAL (V_NAME_F1 (acos), 0.5, 1.0, 50000) 111 PL_TEST_INTERVAL (V_NAME_F1 (acos), 1.0, 0x1p11, 50000) 112 PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p11, inf, 20000) 113 PL_TEST_INTERVAL (V_NAME_F1 (acos), -0, -inf, 20000) 114