1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-precision vector erf(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, 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
12*f3087befSAndrew Turner static const struct data
13*f3087befSAndrew Turner {
14*f3087befSAndrew Turner double third;
15*f3087befSAndrew Turner double tenth, two_over_five, two_over_fifteen;
16*f3087befSAndrew Turner double two_over_nine, two_over_fortyfive;
17*f3087befSAndrew Turner double max, shift;
18*f3087befSAndrew Turner } data = {
19*f3087befSAndrew Turner .third = 0x1.5555555555556p-2, /* used to compute 2/3 and 1/6 too. */
20*f3087befSAndrew Turner .two_over_fifteen = 0x1.1111111111111p-3,
21*f3087befSAndrew Turner .tenth = -0x1.999999999999ap-4,
22*f3087befSAndrew Turner .two_over_five = -0x1.999999999999ap-2,
23*f3087befSAndrew Turner .two_over_nine = -0x1.c71c71c71c71cp-3,
24*f3087befSAndrew Turner .two_over_fortyfive = 0x1.6c16c16c16c17p-5,
25*f3087befSAndrew Turner .max = 5.9921875, /* 6 - 1/128. */
26*f3087befSAndrew Turner .shift = 0x1p45,
27*f3087befSAndrew Turner };
28*f3087befSAndrew Turner
29*f3087befSAndrew Turner #define SignMask (0x8000000000000000)
30*f3087befSAndrew Turner
31*f3087befSAndrew Turner /* Double-precision implementation of vector erf(x).
32*f3087befSAndrew Turner Approximation based on series expansion near x rounded to
33*f3087befSAndrew Turner nearest multiple of 1/128.
34*f3087befSAndrew Turner Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
35*f3087befSAndrew Turner erf(x) ~ erf(r) + scale * d * [
36*f3087befSAndrew Turner + 1
37*f3087befSAndrew Turner - r d
38*f3087befSAndrew Turner + 1/3 (2 r^2 - 1) d^2
39*f3087befSAndrew Turner - 1/6 (r (2 r^2 - 3)) d^3
40*f3087befSAndrew Turner + 1/30 (4 r^4 - 12 r^2 + 3) d^4
41*f3087befSAndrew Turner - 1/90 (4 r^4 - 20 r^2 + 15) d^5
42*f3087befSAndrew Turner ]
43*f3087befSAndrew Turner
44*f3087befSAndrew Turner Maximum measure error: 2.29 ULP
45*f3087befSAndrew Turner _ZGVsMxv_erf(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8
46*f3087befSAndrew Turner want -0x1.20dd59132ebafp-8. */
SV_NAME_D1(erf)47*f3087befSAndrew Turner svfloat64_t SV_NAME_D1 (erf) (svfloat64_t x, const svbool_t pg)
48*f3087befSAndrew Turner {
49*f3087befSAndrew Turner const struct data *dat = ptr_barrier (&data);
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner /* |x| >= 6.0 - 1/128. Opposite conditions except none of them catch NaNs so
52*f3087befSAndrew Turner they can be used in lookup and BSLs to yield the expected results. */
53*f3087befSAndrew Turner svbool_t a_ge_max = svacge (pg, x, dat->max);
54*f3087befSAndrew Turner svbool_t a_lt_max = svaclt (pg, x, dat->max);
55*f3087befSAndrew Turner
56*f3087befSAndrew Turner /* Set r to multiple of 1/128 nearest to |x|. */
57*f3087befSAndrew Turner svfloat64_t a = svabs_x (pg, x);
58*f3087befSAndrew Turner svfloat64_t shift = sv_f64 (dat->shift);
59*f3087befSAndrew Turner svfloat64_t z = svadd_x (pg, a, shift);
60*f3087befSAndrew Turner svuint64_t i = svand_x (pg, svreinterpret_u64 (z), 0xfff);
61*f3087befSAndrew Turner i = svadd_x (pg, i, i);
62*f3087befSAndrew Turner
63*f3087befSAndrew Turner /* Lookup without shortcut for small values but with predicate to avoid
64*f3087befSAndrew Turner segfault for large values and NaNs. */
65*f3087befSAndrew Turner svfloat64_t r = svsub_x (pg, z, shift);
66*f3087befSAndrew Turner svfloat64_t erfr
67*f3087befSAndrew Turner = svld1_gather_index (a_lt_max, &__v_erf_data.tab[0].erf, i);
68*f3087befSAndrew Turner svfloat64_t scale
69*f3087befSAndrew Turner = svld1_gather_index (a_lt_max, &__v_erf_data.tab[0].scale, i);
70*f3087befSAndrew Turner
71*f3087befSAndrew Turner /* erf(x) ~ erf(r) + scale * d * poly (r, d). */
72*f3087befSAndrew Turner svfloat64_t d = svsub_x (pg, a, r);
73*f3087befSAndrew Turner svfloat64_t d2 = svmul_x (pg, d, d);
74*f3087befSAndrew Turner svfloat64_t r2 = svmul_x (pg, r, r);
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner /* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */
77*f3087befSAndrew Turner svfloat64_t p1 = r;
78*f3087befSAndrew Turner svfloat64_t third = sv_f64 (dat->third);
79*f3087befSAndrew Turner svfloat64_t twothird = svmul_x (pg, third, 2.0);
80*f3087befSAndrew Turner svfloat64_t sixth = svmul_x (pg, third, 0.5);
81*f3087befSAndrew Turner svfloat64_t p2 = svmls_x (pg, third, r2, twothird);
82*f3087befSAndrew Turner svfloat64_t p3 = svmad_x (pg, r2, third, -0.5);
83*f3087befSAndrew Turner p3 = svmul_x (pg, r, p3);
84*f3087befSAndrew Turner svfloat64_t p4
85*f3087befSAndrew Turner = svmla_x (pg, sv_f64 (dat->two_over_five), r2, dat->two_over_fifteen);
86*f3087befSAndrew Turner p4 = svmls_x (pg, sv_f64 (dat->tenth), r2, p4);
87*f3087befSAndrew Turner svfloat64_t p5
88*f3087befSAndrew Turner = svmla_x (pg, sv_f64 (dat->two_over_nine), r2, dat->two_over_fortyfive);
89*f3087befSAndrew Turner p5 = svmla_x (pg, sixth, r2, p5);
90*f3087befSAndrew Turner p5 = svmul_x (pg, r, p5);
91*f3087befSAndrew Turner
92*f3087befSAndrew Turner svfloat64_t p34 = svmla_x (pg, p3, d, p4);
93*f3087befSAndrew Turner svfloat64_t p12 = svmla_x (pg, p1, d, p2);
94*f3087befSAndrew Turner svfloat64_t y = svmla_x (pg, p34, d2, p5);
95*f3087befSAndrew Turner y = svmla_x (pg, p12, d2, y);
96*f3087befSAndrew Turner
97*f3087befSAndrew Turner y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y));
98*f3087befSAndrew Turner
99*f3087befSAndrew Turner /* Solves the |x| = inf and NaN cases. */
100*f3087befSAndrew Turner y = svsel (a_ge_max, sv_f64 (1.0), y);
101*f3087befSAndrew Turner
102*f3087befSAndrew Turner /* Copy sign. */
103*f3087befSAndrew Turner svuint64_t ix = svreinterpret_u64 (x);
104*f3087befSAndrew Turner svuint64_t iy = svreinterpret_u64 (y);
105*f3087befSAndrew Turner svuint64_t sign = svand_x (pg, ix, SignMask);
106*f3087befSAndrew Turner return svreinterpret_f64 (svorr_x (pg, sign, iy));
107*f3087befSAndrew Turner }
108*f3087befSAndrew Turner
109*f3087befSAndrew Turner TEST_SIG (SV, D, 1, erf, -6.0, 6.0)
110*f3087befSAndrew Turner TEST_ULP (SV_NAME_D1 (erf), 1.79)
111*f3087befSAndrew Turner TEST_DISABLE_FENV (SV_NAME_D1 (erf))
112*f3087befSAndrew Turner TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 0, 5.9921875, 40000)
113*f3087befSAndrew Turner TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 5.9921875, inf, 40000)
114*f3087befSAndrew Turner TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 0, inf, 4000)
115*f3087befSAndrew Turner CLOSE_SVE_ATTR
116