1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision sinh(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2022-2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "math_config.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define AbsMask 0x7fffffffffffffff
13*412f47f9SXin Li #define Half 0x3fe0000000000000
14*412f47f9SXin Li #define OFlowBound \
15*412f47f9SXin Li 0x40862e42fefa39f0 /* 0x1.62e42fefa39fp+9, above which using expm1 results \
16*412f47f9SXin Li in NaN. */
17*412f47f9SXin Li
18*412f47f9SXin Li double
19*412f47f9SXin Li __exp_dd (double, double);
20*412f47f9SXin Li
21*412f47f9SXin Li /* Approximation for double-precision sinh(x) using expm1.
22*412f47f9SXin Li sinh(x) = (exp(x) - exp(-x)) / 2.
23*412f47f9SXin Li The greatest observed error is 2.57 ULP:
24*412f47f9SXin Li __v_sinh(0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2
25*412f47f9SXin Li want 0x1.ab34e59d678d9p-2. */
26*412f47f9SXin Li double
sinh(double x)27*412f47f9SXin Li sinh (double x)
28*412f47f9SXin Li {
29*412f47f9SXin Li uint64_t ix = asuint64 (x);
30*412f47f9SXin Li uint64_t iax = ix & AbsMask;
31*412f47f9SXin Li double ax = asdouble (iax);
32*412f47f9SXin Li uint64_t sign = ix & ~AbsMask;
33*412f47f9SXin Li double halfsign = asdouble (Half | sign);
34*412f47f9SXin Li
35*412f47f9SXin Li if (unlikely (iax >= OFlowBound))
36*412f47f9SXin Li {
37*412f47f9SXin Li /* Special values and overflow. */
38*412f47f9SXin Li if (unlikely (iax > 0x7ff0000000000000))
39*412f47f9SXin Li return __math_invalidf (x);
40*412f47f9SXin Li /* expm1 overflows a little before sinh. We have to fill this
41*412f47f9SXin Li gap by using a different algorithm, in this case we use a
42*412f47f9SXin Li double-precision exp helper. For large x sinh(x) is dominated
43*412f47f9SXin Li by exp(x), however we cannot compute exp without overflow
44*412f47f9SXin Li either. We use the identity: exp(a) = (exp(a / 2)) ^ 2
45*412f47f9SXin Li to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0
46*412f47f9SXin Li ~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. */
47*412f47f9SXin Li double e = __exp_dd (ax / 2, 0);
48*412f47f9SXin Li return (e * halfsign) * e;
49*412f47f9SXin Li }
50*412f47f9SXin Li
51*412f47f9SXin Li /* Use expm1f to retain acceptable precision for small numbers.
52*412f47f9SXin Li Let t = e^(|x|) - 1. */
53*412f47f9SXin Li double t = expm1 (ax);
54*412f47f9SXin Li /* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0
55*412f47f9SXin Li (t + t / (t + 1)) / -2 for x < 0. */
56*412f47f9SXin Li return (t + t / (t + 1)) * halfsign;
57*412f47f9SXin Li }
58*412f47f9SXin Li
59*412f47f9SXin Li PL_SIG (S, D, 1, sinh, -10.0, 10.0)
60*412f47f9SXin Li PL_TEST_ULP (sinh, 2.08)
61*412f47f9SXin Li PL_TEST_SYM_INTERVAL (sinh, 0, 0x1p-51, 100)
62*412f47f9SXin Li PL_TEST_SYM_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000)
63*412f47f9SXin Li PL_TEST_SYM_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000)
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