1 /*
2 * Single-precision vector erf(x) function.
3 *
4 * Copyright (c) 2023-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "sv_math.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 static const struct data
13 {
14 float min, max, scale, shift, third;
15 } data = {
16 .min = 0x1.cp-7f, /* 1/64 - 1/512. */
17 .max = 3.9375, /* 4 - 8/128. */
18 .scale = 0x1.20dd76p+0f, /* 2/sqrt(pi). */
19 .shift = 0x1p16f,
20 .third = 0x1.555556p-2f, /* 1/3. */
21 };
22
23 #define SignMask (0x80000000)
24
25 /* Single-precision implementation of vector erf(x).
26 Approximation based on series expansion near x rounded to
27 nearest multiple of 1/128.
28 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
29
30 erf(x) ~ erf(r) + scale * d * [1 - r * d - 1/3 * d^2]
31
32 Values of erf(r) and scale are read from lookup tables.
33 For |x| < 0x1.cp-7, the algorithm sets r = 0, erf(r) = 0, and scale = 2 /
34 sqrt(pi), so it simply boils down to a Taylor series expansion near 0. For
35 |x| > 3.9375, erf(|x|) rounds to 1.0f.
36
37 Maximum error on each interval:
38 - [0, 0x1.cp-7]: 1.93 ULP
39 _ZGVsMxv_erff(0x1.c373e6p-9) got 0x1.fd686cp-9 want 0x1.fd6868p-9
40 - [0x1.cp-7, 4.0]: 1.26 ULP
41 _ZGVsMxv_erff(0x1.1d002ep+0) got 0x1.c4eb9ap-1 want 0x1.c4eb98p-1. */
SV_NAME_F1(erf)42 svfloat32_t SV_NAME_F1 (erf) (svfloat32_t x, const svbool_t pg)
43 {
44 const struct data *dat = ptr_barrier (&data);
45
46 /* |x| > 1/64 - 1/512. */
47 svbool_t a_gt_min = svacgt (pg, x, dat->min);
48
49 /* |x| >= 4.0 - 8/128. */
50 svbool_t a_ge_max = svacge (pg, x, dat->max);
51 svfloat32_t a = svabs_x (pg, x);
52
53 svfloat32_t shift = sv_f32 (dat->shift);
54 svfloat32_t z = svadd_x (pg, a, shift);
55 svuint32_t i = svand_x (pg, svreinterpret_u32 (z), 0xfff);
56 i = svadd_x (pg, i, i);
57
58 /* r and erf(r) set to 0 for |x| below min. */
59 svfloat32_t r = svsub_z (a_gt_min, z, shift);
60 svfloat32_t erfr
61 = svld1_gather_index (a_gt_min, &__v_erff_data.tab[0].erf, i);
62
63 /* scale set to 2/sqrt(pi) for |x| below min. */
64 svfloat32_t scale
65 = svld1_gather_index (a_gt_min, &__v_erff_data.tab[0].scale, i);
66 scale = svsel (a_gt_min, scale, sv_f32 (dat->scale));
67
68 /* erf(x) ~ erf(r) + scale * d * (1 - r * d + 1/3 * d^2). */
69 svfloat32_t d = svsub_x (pg, a, r);
70 svfloat32_t d2 = svmul_x (pg, d, d);
71 svfloat32_t y = svmla_x (pg, r, d, dat->third);
72 y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y));
73
74 /* Solves the |x| = inf case. */
75 y = svsel (a_ge_max, sv_f32 (1.0f), y);
76
77 /* Copy sign. */
78 svuint32_t ix = svreinterpret_u32 (x);
79 svuint32_t iy = svreinterpret_u32 (y);
80 svuint32_t sign = svand_x (pg, ix, SignMask);
81 return svreinterpret_f32 (svorr_x (pg, sign, iy));
82 }
83
84 TEST_SIG (SV, F, 1, erf, -4.0, 4.0)
85 TEST_ULP (SV_NAME_F1 (erf), 1.43)
86 TEST_DISABLE_FENV (SV_NAME_F1 (erf))
87 TEST_SYM_INTERVAL (SV_NAME_F1 (erf), 0, 0x1.cp-7, 40000)
88 TEST_SYM_INTERVAL (SV_NAME_F1 (erf), 0x1.cp-7, 3.9375, 40000)
89 TEST_SYM_INTERVAL (SV_NAME_F1 (erf), 3.9375, inf, 40000)
90 TEST_SYM_INTERVAL (SV_NAME_F1 (erf), 0, inf, 4000)
91 CLOSE_SVE_ATTR
92