1 /* 2 * Double-precision vector atan(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 [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 #define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30). */ 35 #define BigBound 0x4340000000000000 /* asuint64(0x1p53). */ 36 37 /* Fast implementation of vector atan. 38 Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using 39 z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps: 40 _ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1 41 want 0x1.9225645bdd7c3p-1. */ 42 float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x) 43 { 44 const struct data *d = ptr_barrier (&data); 45 46 /* Small cases, infs and nans are supported by our approximation technique, 47 but do not set fenv flags correctly. Only trigger special case if we need 48 fenv. */ 49 uint64x2_t ix = vreinterpretq_u64_f64 (x); 50 uint64x2_t sign = vandq_u64 (ix, SignMask); 51 52 #if WANT_SIMD_EXCEPT 53 uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000)); 54 uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)), 55 v_u64 (BigBound - TinyBound)); 56 /* If any lane is special, fall back to the scalar routine for all lanes. */ 57 if (unlikely (v_any_u64 (special))) 58 return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1)); 59 #endif 60 61 /* Argument reduction: 62 y := arctan(x) for x < 1 63 y := pi/2 + arctan(-1/x) for x > 1 64 Hence, use z=-1/a if x>=1, otherwise z=a. */ 65 uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0)); 66 /* Avoid dependency in abs(x) in division (and comparison). */ 67 float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x); 68 float64x2_t shift = vreinterpretq_f64_u64 ( 69 vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2))); 70 /* Use absolute value only when needed (odd powers of z). */ 71 float64x2_t az = vbslq_f64 ( 72 SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z); 73 74 /* Calculate the polynomial approximation. 75 Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of 76 full scheme to avoid underflow in x^16. 77 The order 19 polynomial P approximates 78 (atan(sqrt(x))-sqrt(x))/x^(3/2). */ 79 float64x2_t z2 = vmulq_f64 (z, z); 80 float64x2_t x2 = vmulq_f64 (z2, z2); 81 float64x2_t x4 = vmulq_f64 (x2, x2); 82 float64x2_t x8 = vmulq_f64 (x4, x4); 83 float64x2_t y 84 = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, d->poly), 85 v_estrin_11_f64 (z2, x2, x4, x8, d->poly + 8), x8); 86 87 /* Finalize. y = shift + z + z^3 * P(z^2). */ 88 y = vfmaq_f64 (az, y, vmulq_f64 (z2, az)); 89 y = vaddq_f64 (y, shift); 90 91 /* y = atan(x) if x>0, -atan(-x) otherwise. */ 92 y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign)); 93 return y; 94 } 95 96 PL_SIG (V, D, 1, atan, -10.0, 10.0) 97 PL_TEST_ULP (V_NAME_D1 (atan), 1.78) 98 PL_TEST_EXPECT_FENV (V_NAME_D1 (atan), WANT_SIMD_EXCEPT) 99 PL_TEST_INTERVAL (V_NAME_D1 (atan), 0, 0x1p-30, 10000) 100 PL_TEST_INTERVAL (V_NAME_D1 (atan), -0, -0x1p-30, 1000) 101 PL_TEST_INTERVAL (V_NAME_D1 (atan), 0x1p-30, 0x1p53, 900000) 102 PL_TEST_INTERVAL (V_NAME_D1 (atan), -0x1p-30, -0x1p53, 90000) 103 PL_TEST_INTERVAL (V_NAME_D1 (atan), 0x1p53, inf, 10000) 104 PL_TEST_INTERVAL (V_NAME_D1 (atan), -0x1p53, -inf, 1000) 105