1 /* 2 * Double-precision vector atan2(x) function. 3 * 4 * Copyright (c) 2021-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 #include "poly_advsimd_f64.h" 12 13 static const struct data 14 { 15 float64x2_t pi_over_2; 16 float64x2_t poly[20]; 17 } data = { 18 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on 19 the interval [2**-1022, 1.0]. */ 20 .poly = { V2 (-0x1.5555555555555p-2), V2 (0x1.99999999996c1p-3), 21 V2 (-0x1.2492492478f88p-3), V2 (0x1.c71c71bc3951cp-4), 22 V2 (-0x1.745d160a7e368p-4), V2 (0x1.3b139b6a88ba1p-4), 23 V2 (-0x1.11100ee084227p-4), V2 (0x1.e1d0f9696f63bp-5), 24 V2 (-0x1.aebfe7b418581p-5), V2 (0x1.842dbe9b0d916p-5), 25 V2 (-0x1.5d30140ae5e99p-5), V2 (0x1.338e31eb2fbbcp-5), 26 V2 (-0x1.00e6eece7de8p-5), V2 (0x1.860897b29e5efp-6), 27 V2 (-0x1.0051381722a59p-6), V2 (0x1.14e9dc19a4a4ep-7), 28 V2 (-0x1.d0062b42fe3bfp-9), V2 (0x1.17739e210171ap-10), 29 V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), }, 30 .pi_over_2 = V2 (0x1.921fb54442d18p+0), 31 }; 32 33 #define SignMask v_u64 (0x8000000000000000) 34 35 /* Special cases i.e. 0, infinity, NaN (fall back to scalar calls). */ 36 static float64x2_t VPCS_ATTR NOINLINE 37 special_case (float64x2_t y, float64x2_t x, float64x2_t ret, uint64x2_t cmp) 38 { 39 return v_call2_f64 (atan2, y, x, ret, cmp); 40 } 41 42 /* Returns 1 if input is the bit representation of 0, infinity or nan. */ 43 static inline uint64x2_t 44 zeroinfnan (uint64x2_t i) 45 { 46 /* (2 * i - 1) >= (2 * asuint64 (INFINITY) - 1). */ 47 return vcgeq_u64 (vsubq_u64 (vaddq_u64 (i, i), v_u64 (1)), 48 v_u64 (2 * asuint64 (INFINITY) - 1)); 49 } 50 51 /* Fast implementation of vector atan2. 52 Maximum observed error is 2.8 ulps: 53 _ZGVnN2vv_atan2 (0x1.9651a429a859ap+5, 0x1.953075f4ee26p+5) 54 got 0x1.92d628ab678ccp-1 55 want 0x1.92d628ab678cfp-1. */ 56 float64x2_t VPCS_ATTR V_NAME_D2 (atan2) (float64x2_t y, float64x2_t x) 57 { 58 const struct data *data_ptr = ptr_barrier (&data); 59 60 uint64x2_t ix = vreinterpretq_u64_f64 (x); 61 uint64x2_t iy = vreinterpretq_u64_f64 (y); 62 63 uint64x2_t special_cases = vorrq_u64 (zeroinfnan (ix), zeroinfnan (iy)); 64 65 uint64x2_t sign_x = vandq_u64 (ix, SignMask); 66 uint64x2_t sign_y = vandq_u64 (iy, SignMask); 67 uint64x2_t sign_xy = veorq_u64 (sign_x, sign_y); 68 69 float64x2_t ax = vabsq_f64 (x); 70 float64x2_t ay = vabsq_f64 (y); 71 72 uint64x2_t pred_xlt0 = vcltzq_f64 (x); 73 uint64x2_t pred_aygtax = vcgtq_f64 (ay, ax); 74 75 /* Set up z for call to atan. */ 76 float64x2_t n = vbslq_f64 (pred_aygtax, vnegq_f64 (ax), ay); 77 float64x2_t d = vbslq_f64 (pred_aygtax, ay, ax); 78 float64x2_t z = vdivq_f64 (n, d); 79 80 /* Work out the correct shift. */ 81 float64x2_t shift = vreinterpretq_f64_u64 ( 82 vandq_u64 (pred_xlt0, vreinterpretq_u64_f64 (v_f64 (-2.0)))); 83 shift = vbslq_f64 (pred_aygtax, vaddq_f64 (shift, v_f64 (1.0)), shift); 84 shift = vmulq_f64 (shift, data_ptr->pi_over_2); 85 86 /* Calculate the polynomial approximation. 87 Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of 88 full scheme to avoid underflow in x^16. 89 The order 19 polynomial P approximates 90 (atan(sqrt(x))-sqrt(x))/x^(3/2). */ 91 float64x2_t z2 = vmulq_f64 (z, z); 92 float64x2_t x2 = vmulq_f64 (z2, z2); 93 float64x2_t x4 = vmulq_f64 (x2, x2); 94 float64x2_t x8 = vmulq_f64 (x4, x4); 95 float64x2_t ret 96 = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, data_ptr->poly), 97 v_estrin_11_f64 (z2, x2, x4, x8, data_ptr->poly + 8), x8); 98 99 /* Finalize. y = shift + z + z^3 * P(z^2). */ 100 ret = vfmaq_f64 (z, ret, vmulq_f64 (z2, z)); 101 ret = vaddq_f64 (ret, shift); 102 103 /* Account for the sign of x and y. */ 104 ret = vreinterpretq_f64_u64 ( 105 veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy)); 106 107 if (unlikely (v_any_u64 (special_cases))) 108 return special_case (y, x, ret, special_cases); 109 110 return ret; 111 } 112 113 /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */ 114 PL_SIG (V, D, 2, atan2) 115 // TODO tighten this once __v_atan2 is fixed 116 PL_TEST_ULP (V_NAME_D2 (atan2), 2.9) 117 PL_TEST_INTERVAL (V_NAME_D2 (atan2), -10.0, 10.0, 50000) 118 PL_TEST_INTERVAL (V_NAME_D2 (atan2), -1.0, 1.0, 40000) 119 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 0.0, 1.0, 40000) 120 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 1.0, 100.0, 40000) 121 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 1e6, 1e32, 40000) 122