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
18double
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.  */
26double
27sinh (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
59PL_SIG (S, D, 1, sinh, -10.0, 10.0)
60PL_TEST_ULP (sinh, 2.08)
61PL_TEST_SYM_INTERVAL (sinh, 0, 0x1p-51, 100)
62PL_TEST_SYM_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000)
63PL_TEST_SYM_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000)
64