1 /*
2 * Double-precision 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 "math_config.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11
12 #define Shift 0x1p45
13 #define P20 0x1.5555555555555p-2 /* 1/3. */
14 #define P21 0x1.5555555555555p-1 /* 2/3. */
15
16 #define P40 0x1.999999999999ap-4 /* 1/10. */
17 #define P41 0x1.999999999999ap-2 /* 2/5. */
18 #define P42 0x1.11111111111111p-3 /* 2/15. */
19
20 #define P50 0x1.5555555555555p-3 /* 1/6. */
21 #define P51 0x1.c71c71c71c71cp-3 /* 2/9. */
22 #define P52 0x1.6c16c16c16c17p-5 /* 2/45. */
23
24 /* Qi = (i+1) / i. */
25 #define Q5 0x1.3333333333333p0
26 #define Q6 0x1.2aaaaaaaaaaabp0
27 #define Q7 0x1.2492492492492p0
28 #define Q8 0x1.2p0
29 #define Q9 0x1.1c71c71c71c72p0
30
31 /* Ri = -2 * i / ((i+1)*(i+2)). */
32 #define R5 -0x1.e79e79e79e79ep-3
33 #define R6 -0x1.b6db6db6db6dbp-3
34 #define R7 -0x1.8e38e38e38e39p-3
35 #define R8 -0x1.6c16c16c16c17p-3
36 #define R9 -0x1.4f2094f2094f2p-3
37
38 /* Fast erfc approximation based on series expansion near x rounded to
39 nearest multiple of 1/128.
40 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
41
42 erfc(x) ~ erfc(r) - scale * d * poly(r, d), with
43
44 poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3
45 + (2/15 r^4 - 2/5 r^2 + 1/10) d^4
46 - r * (2/45 r^4 - 2/9 r^2 + 1/6) d^5
47 + p6(r) d^6 + ... + p10(r) d^10
48
49 Polynomials p6(r) to p10(r) are computed using recurrence relation
50
51 2(i+1)p_i + 2r(i+2)p_{i+1} + (i+2)(i+3)p_{i+2} = 0,
52 with p0 = 1, and p1(r) = -r.
53
54 Values of erfc(r) and scale(r) are read from lookup tables. Stored values
55 are scaled to avoid hitting the subnormal range.
56
57 Note that for x < 0, erfc(x) = 2.0 - erfc(-x).
58
59 Maximum measured error: 1.71 ULP
60 erfc(0x1.46cfe976733p+4) got 0x1.e15fcbea3e7afp-608
61 want 0x1.e15fcbea3e7adp-608. */
62 double
erfc(double x)63 erfc (double x)
64 {
65 /* Get top words and sign. */
66 uint64_t ix = asuint64 (x);
67 uint64_t ia = ix & 0x7fffffffffffffff;
68 double a = asdouble (ia);
69 uint64_t sign = ix & ~0x7fffffffffffffff;
70
71 /* erfc(nan)=nan, erfc(+inf)=0 and erfc(-inf)=2. */
72 if (unlikely (ia >= 0x7ff0000000000000))
73 return asdouble (sign >> 1) + 1.0 / x; /* Special cases. */
74
75 /* Return early for large enough negative values. */
76 if (x < -6.0)
77 return 2.0;
78
79 /* For |x| < 3487.0/128.0, the following approximation holds. */
80 if (likely (ia < 0x403b3e0000000000))
81 {
82 /* |x| < 0x1p-511 => accurate to 0.5 ULP. */
83 if (unlikely (ia < asuint64 (0x1p-511)))
84 return 1.0 - x;
85
86 /* Lookup erfc(r) and scale(r) in tables, e.g. set erfc(r) to 1 and scale
87 to 2/sqrt(pi), when x reduced to r = 0. */
88 double z = a + Shift;
89 uint64_t i = asuint64 (z);
90 double r = z - Shift;
91 /* These values are scaled by 2^128. */
92 double erfcr = __erfc_data.tab[i].erfc;
93 double scale = __erfc_data.tab[i].scale;
94
95 /* erfc(x) ~ erfc(r) - scale * d * poly (r, d). */
96 double d = a - r;
97 double d2 = d * d;
98 double r2 = r * r;
99 /* Compute p_i as a regular (low-order) polynomial. */
100 double p1 = -r;
101 double p2 = fma (P21, r2, -P20);
102 double p3 = -r * fma (P20, r2, -0.5);
103 double p4 = fma (fma (P42, r2, -P41), r2, P40);
104 double p5 = -r * fma (fma (P52, r2, -P51), r2, P50);
105 /* Compute p_i using recurrence relation:
106 p_{i+2} = (p_i + r * Q_{i+1} * p_{i+1}) * R_{i+1}. */
107 double p6 = fma (Q5 * r, p5, p4) * R5;
108 double p7 = fma (Q6 * r, p6, p5) * R6;
109 double p8 = fma (Q7 * r, p7, p6) * R7;
110 double p9 = fma (Q8 * r, p8, p7) * R8;
111 double p10 = fma (Q9 * r, p9, p8) * R9;
112 /* Compute polynomial in d using pairwise Horner scheme. */
113 double p90 = fma (p10, d, p9);
114 double p78 = fma (p8, d, p7);
115 double p56 = fma (p6, d, p5);
116 double p34 = fma (p4, d, p3);
117 double p12 = fma (p2, d, p1);
118 double y = fma (p90, d2, p78);
119 y = fma (y, d2, p56);
120 y = fma (y, d2, p34);
121 y = fma (y, d2, p12);
122
123 y = fma (-fma (y, d2, d), scale, erfcr);
124
125 /* Handle sign and scale back in a single fma. */
126 double off = asdouble (sign >> 1);
127 double fac = asdouble (asuint64 (0x1p-128) | sign);
128 y = fma (y, fac, off);
129
130 if (unlikely (x > 26.0))
131 {
132 /* The underflow exception needs to be signaled explicitly when
133 result gets into the subnormal range. */
134 if (unlikely (y < 0x1p-1022))
135 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
136 /* Set errno to ERANGE if result rounds to 0. */
137 return __math_check_uflow (y);
138 }
139
140 return y;
141 }
142 /* Above the threshold (x > 3487.0/128.0) erfc is constant and needs to raise
143 underflow exception for positive x. */
144 return __math_uflow (0);
145 }
146
147 PL_SIG (S, D, 1, erfc, -6.0, 28.0)
148 PL_TEST_ULP (erfc, 1.21)
149 PL_TEST_SYM_INTERVAL (erfc, 0, 0x1p-26, 40000)
150 PL_TEST_INTERVAL (erfc, 0x1p-26, 28.0, 100000)
151 PL_TEST_INTERVAL (erfc, -0x1p-26, -6.0, 100000)
152 PL_TEST_INTERVAL (erfc, 28.0, inf, 40000)
153 PL_TEST_INTERVAL (erfc, -6.0, -inf, 40000)
154