1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-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 "sv_math.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner #include "sv_poly_f64.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner static const struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner float64_t poly[20];
16*f3087befSAndrew Turner float64_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**-1022, 1.0]. */
20*f3087befSAndrew Turner .poly = { -0x1.5555555555555p-2, 0x1.99999999996c1p-3, -0x1.2492492478f88p-3,
21*f3087befSAndrew Turner 0x1.c71c71bc3951cp-4, -0x1.745d160a7e368p-4, 0x1.3b139b6a88ba1p-4,
22*f3087befSAndrew Turner -0x1.11100ee084227p-4, 0x1.e1d0f9696f63bp-5, -0x1.aebfe7b418581p-5,
23*f3087befSAndrew Turner 0x1.842dbe9b0d916p-5, -0x1.5d30140ae5e99p-5, 0x1.338e31eb2fbbcp-5,
24*f3087befSAndrew Turner -0x1.00e6eece7de8p-5, 0x1.860897b29e5efp-6, -0x1.0051381722a59p-6,
25*f3087befSAndrew Turner 0x1.14e9dc19a4a4ep-7, -0x1.d0062b42fe3bfp-9, 0x1.17739e210171ap-10,
26*f3087befSAndrew Turner -0x1.ab24da7be7402p-13, 0x1.358851160a528p-16, },
27*f3087befSAndrew Turner .pi_over_2 = 0x1.921fb54442d18p+0,
28*f3087befSAndrew Turner };
29*f3087befSAndrew Turner
30*f3087befSAndrew Turner /* Useful constants. */
31*f3087befSAndrew Turner #define SignMask (0x8000000000000000)
32*f3087befSAndrew Turner
33*f3087befSAndrew Turner /* Fast implementation of SVE atan.
34*f3087befSAndrew Turner Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
35*f3087befSAndrew Turner z=1/x and shift = pi/2. Largest errors are close to 1. The maximum observed
36*f3087befSAndrew Turner error is 2.27 ulps:
37*f3087befSAndrew Turner _ZGVsMxv_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
38*f3087befSAndrew Turner want 0x1.9225645bdd7c3p-1. */
SV_NAME_D1(atan)39*f3087befSAndrew Turner svfloat64_t SV_NAME_D1 (atan) (svfloat64_t x, const svbool_t pg)
40*f3087befSAndrew Turner {
41*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
42*f3087befSAndrew Turner
43*f3087befSAndrew Turner /* No need to trigger special case. Small cases, infs and nans
44*f3087befSAndrew Turner are supported by our approximation technique. */
45*f3087befSAndrew Turner svuint64_t ix = svreinterpret_u64 (x);
46*f3087befSAndrew Turner svuint64_t sign = svand_x (pg, ix, SignMask);
47*f3087befSAndrew Turner
48*f3087befSAndrew Turner /* Argument reduction:
49*f3087befSAndrew Turner y := arctan(x) for x < 1
50*f3087befSAndrew Turner y := pi/2 + arctan(-1/x) for x > 1
51*f3087befSAndrew Turner Hence, use z=-1/a if x>=1, otherwise z=a. */
52*f3087befSAndrew Turner svbool_t red = svacgt (pg, x, 1.0);
53*f3087befSAndrew Turner /* Avoid dependency in abs(x) in division (and comparison). */
54*f3087befSAndrew Turner svfloat64_t z = svsel (red, svdivr_x (pg, x, 1.0), x);
55*f3087befSAndrew Turner /* Use absolute value only when needed (odd powers of z). */
56*f3087befSAndrew Turner svfloat64_t az = svabs_x (pg, z);
57*f3087befSAndrew Turner az = svneg_m (az, red, az);
58*f3087befSAndrew Turner
59*f3087befSAndrew Turner /* Use split Estrin scheme for P(z^2) with deg(P)=19. */
60*f3087befSAndrew Turner svfloat64_t z2 = svmul_x (pg, z, z);
61*f3087befSAndrew Turner svfloat64_t x2 = svmul_x (pg, z2, z2);
62*f3087befSAndrew Turner svfloat64_t x4 = svmul_x (pg, x2, x2);
63*f3087befSAndrew Turner svfloat64_t x8 = svmul_x (pg, x4, x4);
64*f3087befSAndrew Turner
65*f3087befSAndrew Turner svfloat64_t y
66*f3087befSAndrew Turner = svmla_x (pg, sv_estrin_7_f64_x (pg, z2, x2, x4, d->poly),
67*f3087befSAndrew Turner sv_estrin_11_f64_x (pg, z2, x2, x4, x8, d->poly + 8), x8);
68*f3087befSAndrew Turner
69*f3087befSAndrew Turner /* y = shift + z + z^3 * P(z^2). */
70*f3087befSAndrew Turner svfloat64_t z3 = svmul_x (pg, z2, az);
71*f3087befSAndrew Turner y = svmla_x (pg, az, z3, y);
72*f3087befSAndrew Turner
73*f3087befSAndrew Turner /* Apply shift as indicated by `red` predicate. */
74*f3087befSAndrew Turner y = svadd_m (red, y, d->pi_over_2);
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner /* y = atan(x) if x>0, -atan(-x) otherwise. */
77*f3087befSAndrew Turner y = svreinterpret_f64 (sveor_x (pg, svreinterpret_u64 (y), sign));
78*f3087befSAndrew Turner
79*f3087befSAndrew Turner return y;
80*f3087befSAndrew Turner }
81*f3087befSAndrew Turner
82*f3087befSAndrew Turner TEST_SIG (SV, D, 1, atan, -3.1, 3.1)
83*f3087befSAndrew Turner TEST_ULP (SV_NAME_D1 (atan), 1.78)
84*f3087befSAndrew Turner TEST_DISABLE_FENV (SV_NAME_D1 (atan))
85*f3087befSAndrew Turner TEST_INTERVAL (SV_NAME_D1 (atan), 0.0, 1.0, 40000)
86*f3087befSAndrew Turner TEST_INTERVAL (SV_NAME_D1 (atan), 1.0, 100.0, 40000)
87*f3087befSAndrew Turner TEST_INTERVAL (SV_NAME_D1 (atan), 100, inf, 40000)
88*f3087befSAndrew Turner TEST_INTERVAL (SV_NAME_D1 (atan), -0, -inf, 40000)
89*f3087befSAndrew Turner CLOSE_SVE_ATTR
90