1 /*
2 * Single-precision erfc(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 "math_config.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 #define Shift 0x1p17f
13 #define OneThird 0x1.555556p-2f
14 #define TwoThird 0x1.555556p-1f
15
16 #define TwoOverFifteen 0x1.111112p-3f
17 #define TwoOverFive 0x1.99999ap-2f
18 #define Tenth 0x1.99999ap-4f
19
20 #define SignMask 0x7fffffff
21
22 /* Fast erfcf approximation based on series expansion near x rounded to
23 nearest multiple of 1/64.
24 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
25
26 erfc(x) ~ erfc(r) - scale * d * poly(r, d), with
27
28 poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3
29 + (2/15 r^4 - 2/5 r^2 + 1/10) d^4
30
31 Values of erfc(r) and scale are read from lookup tables. Stored values
32 are scaled to avoid hitting the subnormal range.
33
34 Note that for x < 0, erfc(x) = 2.0 - erfc(-x).
35
36 Maximum error: 1.63 ULP (~1.0 ULP for x < 0.0).
37 erfcf(0x1.1dbf7ap+3) got 0x1.f51212p-120
38 want 0x1.f51216p-120. */
39 float
erfcf(float x)40 erfcf (float x)
41 {
42 /* Get top words and sign. */
43 uint32_t ix = asuint (x);
44 uint32_t ia = ix & SignMask;
45 uint32_t sign = ix & ~SignMask;
46
47 /* |x| < 0x1.0p-26 => accurate to 0.5 ULP (top12(0x1p-26) = 0x328). */
48 if (unlikely (ia < 0x32800000))
49 return 1.0f - x; /* Small case. */
50
51 /* For |x| < 10.0625, the following approximation holds. */
52 if (likely (ia < 0x41210000))
53 {
54 /* Lookup erfc(r) and scale(r) in tables, e.g. set erfc(r) to 1 and scale
55 to 2/sqrt(pi), when x reduced to r = 0. */
56 float a = asfloat (ia);
57 float z = a + Shift;
58 uint32_t i = asuint (z) - asuint (Shift);
59 float r = z - Shift;
60
61 /* These values are scaled by 2^-47. */
62 float erfcr = __v_erfcf_data.tab[i].erfc;
63 float scale = __v_erfcf_data.tab[i].scale;
64
65 /* erfc(x) ~ erfc(r) - scale * d * poly (r, d). */
66 float d = a - r;
67 float d2 = d * d;
68 float r2 = r * r;
69 float p1 = -r;
70 float p2 = fmaf (TwoThird, r2, -OneThird);
71 float p3 = -r * fmaf (OneThird, r2, -0.5f);
72 float p4 = fmaf (fmaf (TwoOverFifteen, r2, -TwoOverFive), r2, Tenth);
73 float y = fmaf (p4, d, p3);
74 y = fmaf (y, d, p2);
75 y = fmaf (y, d, p1);
76 y = fmaf (-fmaf (y, d2, d), scale, erfcr);
77 /* Handle sign and scale back in a single fma. */
78 float off = asfloat (sign >> 1);
79 float fac = asfloat (asuint (0x1p-47f) | sign);
80 y = fmaf (y, fac, off);
81 /* The underflow exception needs to be signaled explicitly when
82 result gets into subormnal range. */
83 if (x >= 0x1.2639cp+3f)
84 force_eval_float (opt_barrier_float (0x1p-123f) * 0x1p-123f);
85 return y;
86 }
87
88 /* erfcf(nan)=nan, erfcf(+inf)=0 and erfcf(-inf)=2. */
89 if (unlikely (ia >= 0x7f800000))
90 return asfloat (sign >> 1) + 1.0f / x; /* Special cases. */
91
92 /* Above this threshold erfcf is constant and needs to raise underflow
93 exception for positive x. */
94 return sign ? 2.0f : __math_uflowf (0);
95 }
96
97 TEST_SIG (S, F, 1, erfc, -4.0, 10.0)
98 TEST_ULP (erfcf, 1.14)
99 TEST_SYM_INTERVAL (erfcf, 0, 0x1p-26, 40000)
100 TEST_INTERVAL (erfcf, 0x1p-26, 10.0625, 40000)
101 TEST_INTERVAL (erfcf, -0x1p-26, -4.0, 40000)
102 TEST_INTERVAL (erfcf, 10.0625, inf, 40000)
103 TEST_INTERVAL (erfcf, -4.0, -inf, 40000)
104