1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Single-precision vector atan(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2021-2024, 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 "test_sig.h" 10*f3087befSAndrew Turner #include "test_defs.h" 11*f3087befSAndrew Turner #include "v_poly_f32.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner static const struct data 14*f3087befSAndrew Turner { 15*f3087befSAndrew Turner float32x4_t poly[8]; 16*f3087befSAndrew Turner float32x4_t pi_over_2; 17*f3087befSAndrew Turner } data = { 18*f3087befSAndrew Turner /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on 19*f3087befSAndrew Turner [2**-128, 1.0]. 20*f3087befSAndrew Turner Generated using fpminimax between FLT_MIN and 1. */ 21*f3087befSAndrew Turner .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f), 22*f3087befSAndrew Turner V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f), 23*f3087befSAndrew Turner V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) }, 24*f3087befSAndrew Turner .pi_over_2 = V4 (0x1.921fb6p+0f), 25*f3087befSAndrew Turner }; 26*f3087befSAndrew Turner 27*f3087befSAndrew Turner #define SignMask v_u32 (0x80000000) 28*f3087befSAndrew Turner 29*f3087befSAndrew Turner #define P(i) d->poly[i] 30*f3087befSAndrew Turner 31*f3087befSAndrew Turner #define TinyBound 0x30800000 /* asuint(0x1p-30). */ 32*f3087befSAndrew Turner #define BigBound 0x4e800000 /* asuint(0x1p30). */ 33*f3087befSAndrew Turner 34*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 35*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE 36*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint32x4_t special) 37*f3087befSAndrew Turner { 38*f3087befSAndrew Turner return v_call_f32 (atanf, x, y, special); 39*f3087befSAndrew Turner } 40*f3087befSAndrew Turner #endif 41*f3087befSAndrew Turner 42*f3087befSAndrew Turner /* Fast implementation of vector atanf based on 43*f3087befSAndrew Turner atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] 44*f3087befSAndrew Turner using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps: 45*f3087befSAndrew Turner _ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */ 46*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x) 47*f3087befSAndrew Turner { 48*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 49*f3087befSAndrew Turner 50*f3087befSAndrew Turner /* Small cases, infs and nans are supported by our approximation technique, 51*f3087befSAndrew Turner but do not set fenv flags correctly. Only trigger special case if we need 52*f3087befSAndrew Turner fenv. */ 53*f3087befSAndrew Turner uint32x4_t ix = vreinterpretq_u32_f32 (x); 54*f3087befSAndrew Turner uint32x4_t sign = vandq_u32 (ix, SignMask); 55*f3087befSAndrew Turner 56*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 57*f3087befSAndrew Turner uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000)); 58*f3087befSAndrew Turner uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)), 59*f3087befSAndrew Turner v_u32 (BigBound - TinyBound)); 60*f3087befSAndrew Turner /* If any lane is special, fall back to the scalar routine for all lanes. */ 61*f3087befSAndrew Turner if (unlikely (v_any_u32 (special))) 62*f3087befSAndrew Turner return special_case (x, x, v_u32 (-1)); 63*f3087befSAndrew Turner #endif 64*f3087befSAndrew Turner 65*f3087befSAndrew Turner /* Argument reduction: 66*f3087befSAndrew Turner y := arctan(x) for x < 1 67*f3087befSAndrew Turner y := pi/2 + arctan(-1/x) for x > 1 68*f3087befSAndrew Turner Hence, use z=-1/a if x>=1, otherwise z=a. */ 69*f3087befSAndrew Turner uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0)); 70*f3087befSAndrew Turner /* Avoid dependency in abs(x) in division (and comparison). */ 71*f3087befSAndrew Turner float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x); 72*f3087befSAndrew Turner float32x4_t shift = vreinterpretq_f32_u32 ( 73*f3087befSAndrew Turner vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2))); 74*f3087befSAndrew Turner /* Use absolute value only when needed (odd powers of z). */ 75*f3087befSAndrew Turner float32x4_t az = vbslq_f32 ( 76*f3087befSAndrew Turner SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z); 77*f3087befSAndrew Turner 78*f3087befSAndrew Turner /* Calculate the polynomial approximation. 79*f3087befSAndrew Turner Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However, 80*f3087befSAndrew Turner a standard implementation using z8 creates spurious underflow 81*f3087befSAndrew Turner in the very last fma (when z^8 is small enough). 82*f3087befSAndrew Turner Therefore, we split the last fma into a mul and an fma. 83*f3087befSAndrew Turner Horner and single-level Estrin have higher errors that exceed 84*f3087befSAndrew Turner threshold. */ 85*f3087befSAndrew Turner float32x4_t z2 = vmulq_f32 (z, z); 86*f3087befSAndrew Turner float32x4_t z4 = vmulq_f32 (z2, z2); 87*f3087befSAndrew Turner 88*f3087befSAndrew Turner float32x4_t y = vfmaq_f32 ( 89*f3087befSAndrew Turner v_pairwise_poly_3_f32 (z2, z4, d->poly), z4, 90*f3087befSAndrew Turner vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4))); 91*f3087befSAndrew Turner 92*f3087befSAndrew Turner /* y = shift + z * P(z^2). */ 93*f3087befSAndrew Turner y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift); 94*f3087befSAndrew Turner 95*f3087befSAndrew Turner /* y = atan(x) if x>0, -atan(-x) otherwise. */ 96*f3087befSAndrew Turner y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign)); 97*f3087befSAndrew Turner 98*f3087befSAndrew Turner return y; 99*f3087befSAndrew Turner } 100*f3087befSAndrew Turner 101*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (atan) 102*f3087befSAndrew Turner 103*f3087befSAndrew Turner TEST_SIG (V, F, 1, atan, -10.0, 10.0) 104*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (atan), 2.5) 105*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (atan), WANT_SIMD_EXCEPT) 106*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0, 0x1p-30, 5000) 107*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p-30, 1, 40000) 108*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 1, 0x1p30, 40000) 109*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p30, inf, 1000) 110