1 /*
2 * Double-precision sinh(x) function.
3 *
4 * Copyright (c) 2022-2023, 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 AbsMask 0x7fffffffffffffff
13 #define Half 0x3fe0000000000000
14 #define OFlowBound \
15 0x40862e42fefa39f0 /* 0x1.62e42fefa39fp+9, above which using expm1 results \
16 in NaN. */
17
18 double
19 __exp_dd (double, double);
20
21 /* Approximation for double-precision sinh(x) using expm1.
22 sinh(x) = (exp(x) - exp(-x)) / 2.
23 The greatest observed error is 2.57 ULP:
24 __v_sinh(0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2
25 want 0x1.ab34e59d678d9p-2. */
26 double
sinh(double x)27 sinh (double x)
28 {
29 uint64_t ix = asuint64 (x);
30 uint64_t iax = ix & AbsMask;
31 double ax = asdouble (iax);
32 uint64_t sign = ix & ~AbsMask;
33 double halfsign = asdouble (Half | sign);
34
35 if (unlikely (iax >= OFlowBound))
36 {
37 /* Special values and overflow. */
38 if (unlikely (iax > 0x7ff0000000000000))
39 return __math_invalidf (x);
40 /* expm1 overflows a little before sinh. We have to fill this
41 gap by using a different algorithm, in this case we use a
42 double-precision exp helper. For large x sinh(x) is dominated
43 by exp(x), however we cannot compute exp without overflow
44 either. We use the identity: exp(a) = (exp(a / 2)) ^ 2
45 to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0
46 ~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. */
47 double e = __exp_dd (ax / 2, 0);
48 return (e * halfsign) * e;
49 }
50
51 /* Use expm1f to retain acceptable precision for small numbers.
52 Let t = e^(|x|) - 1. */
53 double t = expm1 (ax);
54 /* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0
55 (t + t / (t + 1)) / -2 for x < 0. */
56 return (t + t / (t + 1)) * halfsign;
57 }
58
59 PL_SIG (S, D, 1, sinh, -10.0, 10.0)
60 PL_TEST_ULP (sinh, 2.08)
61 PL_TEST_SYM_INTERVAL (sinh, 0, 0x1p-51, 100)
62 PL_TEST_SYM_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000)
63 PL_TEST_SYM_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000)
64