1 /* 2 * Single-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 "sv_math.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 #include "poly_sve_f32.h" 12 13 static const struct data 14 { 15 float32_t poly[8]; 16 float32_t pi_over_2; 17 } data = { 18 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on 19 [2**-128, 1.0]. */ 20 .poly = { -0x1.55555p-2f, 0x1.99935ep-3f, -0x1.24051ep-3f, 0x1.bd7368p-4f, 21 -0x1.491f0ep-4f, 0x1.93a2c0p-5f, -0x1.4c3c60p-6f, 0x1.01fd88p-8f }, 22 .pi_over_2 = 0x1.921fb6p+0f, 23 }; 24 25 #define SignMask (0x80000000) 26 27 /* Fast implementation of SVE atanf based on 28 atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using 29 z=-1/x and shift = pi/2. 30 Largest observed error is 2.9 ULP, close to +/-1.0: 31 _ZGVsMxv_atanf (0x1.0468f6p+0) got -0x1.967f06p-1 32 want -0x1.967fp-1. */ 33 svfloat32_t SV_NAME_F1 (atan) (svfloat32_t x, const svbool_t pg) 34 { 35 const struct data *d = ptr_barrier (&data); 36 37 /* No need to trigger special case. Small cases, infs and nans 38 are supported by our approximation technique. */ 39 svuint32_t ix = svreinterpret_u32 (x); 40 svuint32_t sign = svand_x (pg, ix, SignMask); 41 42 /* Argument reduction: 43 y := arctan(x) for x < 1 44 y := pi/2 + arctan(-1/x) for x > 1 45 Hence, use z=-1/a if x>=1, otherwise z=a. */ 46 svbool_t red = svacgt (pg, x, 1.0f); 47 /* Avoid dependency in abs(x) in division (and comparison). */ 48 svfloat32_t z = svsel (red, svdiv_x (pg, sv_f32 (1.0f), x), x); 49 /* Use absolute value only when needed (odd powers of z). */ 50 svfloat32_t az = svabs_x (pg, z); 51 az = svneg_m (az, red, az); 52 53 /* Use split Estrin scheme for P(z^2) with deg(P)=7. */ 54 svfloat32_t z2 = svmul_x (pg, z, z); 55 svfloat32_t z4 = svmul_x (pg, z2, z2); 56 svfloat32_t z8 = svmul_x (pg, z4, z4); 57 58 svfloat32_t y = sv_estrin_7_f32_x (pg, z2, z4, z8, d->poly); 59 60 /* y = shift + z + z^3 * P(z^2). */ 61 svfloat32_t z3 = svmul_x (pg, z2, az); 62 y = svmla_x (pg, az, z3, y); 63 64 /* Apply shift as indicated by 'red' predicate. */ 65 y = svadd_m (red, y, sv_f32 (d->pi_over_2)); 66 67 /* y = atan(x) if x>0, -atan(-x) otherwise. */ 68 return svreinterpret_f32 (sveor_x (pg, svreinterpret_u32 (y), sign)); 69 } 70 71 PL_SIG (SV, F, 1, atan, -3.1, 3.1) 72 PL_TEST_ULP (SV_NAME_F1 (atan), 2.9) 73 PL_TEST_INTERVAL (SV_NAME_F1 (atan), 0.0, 1.0, 40000) 74 PL_TEST_INTERVAL (SV_NAME_F1 (atan), 1.0, 100.0, 40000) 75 PL_TEST_INTERVAL (SV_NAME_F1 (atan), 100, inf, 40000) 76 PL_TEST_INTERVAL (SV_NAME_F1 (atan), -0, -inf, 40000) 77