1 /*
2 * Double-precision asinh(x) function
3 *
4 * Copyright (c) 2022-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7 #include "mathlib.h"
8 #include "poly_scalar_f64.h"
9 #include "math_config.h"
10 #include "test_sig.h"
11 #include "test_defs.h"
12
13 #define AbsMask 0x7fffffffffffffff
14 #define ExpM26 0x3e50000000000000 /* asuint64(0x1.0p-26). */
15 #define One 0x3ff0000000000000 /* asuint64(1.0). */
16 #define Exp511 0x5fe0000000000000 /* asuint64(0x1.0p511). */
17 #define Ln2 0x1.62e42fefa39efp-1
18
19 /* Scalar double-precision asinh implementation. This routine uses different
20 approaches on different intervals:
21
22 |x| < 2^-26: Return x. Function is exact in this region.
23
24 |x| < 1: Use custom order-17 polynomial. This is least accurate close to 1.
25 The largest observed error in this region is 1.47 ULPs:
26 asinh(0x1.fdfcd00cc1e6ap-1) got 0x1.c1d6bf874019bp-1
27 want 0x1.c1d6bf874019cp-1.
28
29 |x| < 2^511: Upper bound of this region is close to sqrt(DBL_MAX). Calculate
30 the result directly using the definition asinh(x) = ln(x + sqrt(x*x + 1)).
31 The largest observed error in this region is 2.03 ULPs:
32 asinh(-0x1.00094e0f39574p+0) got -0x1.c3508eb6a681ep-1
33 want -0x1.c3508eb6a682p-1.
34
35 |x| >= 2^511: We cannot square x without overflow at a low
36 cost. At very large x, asinh(x) ~= ln(2x). At huge x we cannot
37 even double x without overflow, so calculate this as ln(x) +
38 ln(2). The largest observed error in this region is 0.98 ULPs at many
39 values, for instance:
40 asinh(0x1.5255a4cf10319p+975) got 0x1.52652f4cb26cbp+9
41 want 0x1.52652f4cb26ccp+9. */
42 double
asinh(double x)43 asinh (double x)
44 {
45 uint64_t ix = asuint64 (x);
46 uint64_t ia = ix & AbsMask;
47 double ax = asdouble (ia);
48 uint64_t sign = ix & ~AbsMask;
49
50 if (ia < ExpM26)
51 {
52 return x;
53 }
54
55 if (ia < One)
56 {
57 double x2 = x * x;
58 double z2 = x2 * x2;
59 double z4 = z2 * z2;
60 double z8 = z4 * z4;
61 double p = estrin_17_f64 (x2, z2, z4, z8, z8 * z8, __asinh_data.poly);
62 double y = fma (p, x2 * ax, ax);
63 return asdouble (asuint64 (y) | sign);
64 }
65
66 if (unlikely (ia >= Exp511))
67 {
68 return asdouble (asuint64 (log (ax) + Ln2) | sign);
69 }
70
71 return asdouble (asuint64 (log (ax + sqrt (ax * ax + 1))) | sign);
72 }
73
74 TEST_SIG (S, D, 1, asinh, -10.0, 10.0)
75 TEST_ULP (asinh, 1.54)
76 TEST_INTERVAL (asinh, -0x1p-26, 0x1p-26, 50000)
77 TEST_INTERVAL (asinh, 0x1p-26, 1.0, 40000)
78 TEST_INTERVAL (asinh, -0x1p-26, -1.0, 10000)
79 TEST_INTERVAL (asinh, 1.0, 100.0, 40000)
80 TEST_INTERVAL (asinh, -1.0, -100.0, 10000)
81 TEST_INTERVAL (asinh, 100.0, inf, 50000)
82 TEST_INTERVAL (asinh, -100.0, -inf, 10000)
83