1 /*
2 * Single-precision vector erfc(x) function.
3 *
4 * Copyright (c) 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
12 static const struct data
13 {
14 uint32_t off_idx, off_arr;
15 float max, shift;
16 float third, two_thirds, two_over_fifteen, two_over_five, tenth;
17 } data = {
18 /* Set an offset so the range of the index used for lookup is 644, and it can
19 be clamped using a saturated add. */
20 .off_idx = 0xb7fffd7b, /* 0xffffffff - asuint(shift) - 644. */
21 .off_arr = 0xfffffd7b, /* 0xffffffff - 644. */
22 .max = 10.0625f, /* 644/64. */
23 .shift = 0x1p17f,
24 .third = 0x1.555556p-2f,
25 .two_thirds = 0x1.555556p-1f,
26 .two_over_fifteen = 0x1.111112p-3f,
27 .two_over_five = -0x1.99999ap-2f,
28 .tenth = -0x1.99999ap-4f,
29 };
30
31 #define SignMask 0x80000000
32 #define TableScale 0x28000000 /* 0x1p-47. */
33
34 /* Optimized single-precision vector erfcf(x).
35 Approximation based on series expansion near x rounded to
36 nearest multiple of 1/64.
37 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
38
39 erfc(x) ~ erfc(r) - scale * d * poly(r, d), with
40
41 poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3
42 + (2/15 r^4 - 2/5 r^2 + 1/10) d^4
43
44 Values of erfc(r) and scale are read from lookup tables. Stored values
45 are scaled to avoid hitting the subnormal range.
46
47 Note that for x < 0, erfc(x) = 2.0 - erfc(-x).
48
49 Maximum error: 1.63 ULP (~1.0 ULP for x < 0.0).
50 _ZGVsMxv_erfcf(0x1.1dbf7ap+3) got 0x1.f51212p-120
51 want 0x1.f51216p-120. */
SV_NAME_F1(erfc)52 svfloat32_t SV_NAME_F1 (erfc) (svfloat32_t x, const svbool_t pg)
53 {
54 const struct data *dat = ptr_barrier (&data);
55
56 svfloat32_t a = svabs_x (pg, x);
57
58 /* Clamp input at |x| <= 10.0 + 4/64. */
59 a = svmin_x (pg, a, dat->max);
60
61 /* Reduce x to the nearest multiple of 1/64. */
62 svfloat32_t shift = sv_f32 (dat->shift);
63 svfloat32_t z = svadd_x (pg, a, shift);
64
65 /* Saturate index for the NaN case. */
66 svuint32_t i = svqadd (svreinterpret_u32 (z), dat->off_idx);
67
68 /* Lookup erfc(r) and 2/sqrt(pi)*exp(-r^2) in tables. */
69 i = svmul_x (pg, i, 2);
70 const float32_t *p = &__erfcf_data.tab[0].erfc - 2 * dat->off_arr;
71 svfloat32_t erfcr = svld1_gather_index (pg, p, i);
72 svfloat32_t scale = svld1_gather_index (pg, p + 1, i);
73
74 /* erfc(x) ~ erfc(r) - scale * d * poly(r, d). */
75 svfloat32_t r = svsub_x (pg, z, shift);
76 svfloat32_t d = svsub_x (pg, a, r);
77 svfloat32_t d2 = svmul_x (pg, d, d);
78 svfloat32_t r2 = svmul_x (pg, r, r);
79
80 svfloat32_t coeffs = svld1rq (svptrue_b32 (), &dat->third);
81 svfloat32_t third = svdup_lane (coeffs, 0);
82
83 svfloat32_t p1 = r;
84 svfloat32_t p2 = svmls_lane (third, r2, coeffs, 1);
85 svfloat32_t p3 = svmul_x (pg, r, svmla_lane (sv_f32 (-0.5), r2, coeffs, 0));
86 svfloat32_t p4 = svmla_lane (sv_f32 (dat->two_over_five), r2, coeffs, 2);
87 p4 = svmls_x (pg, sv_f32 (dat->tenth), r2, p4);
88
89 svfloat32_t y = svmla_x (pg, p3, d, p4);
90 y = svmla_x (pg, p2, d, y);
91 y = svmla_x (pg, p1, d, y);
92
93 /* Solves the |x| = inf/nan case. */
94 y = svmls_x (pg, erfcr, scale, svmls_x (pg, d, d2, y));
95
96 /* Offset equals 2.0f if sign, else 0.0f. */
97 svuint32_t sign = svand_x (pg, svreinterpret_u32 (x), SignMask);
98 svfloat32_t off = svreinterpret_f32 (svlsr_x (pg, sign, 1));
99 /* Handle sign and scale back in a single fma. */
100 svfloat32_t fac = svreinterpret_f32 (svorr_x (pg, sign, TableScale));
101
102 return svmla_x (pg, off, fac, y);
103 }
104
105 PL_SIG (SV, F, 1, erfc, -4.0, 10.0)
106 PL_TEST_ULP (SV_NAME_F1 (erfc), 1.14)
107 PL_TEST_SYM_INTERVAL (SV_NAME_F1 (erfc), 0.0, 0x1p-26, 40000)
108 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), 0x1p-26, 10.0625, 40000)
109 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), -0x1p-26, -4.0, 40000)
110 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), 10.0625, inf, 40000)
111 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), -4.0, -inf, 40000)
112