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/*
* Double-precision sinh(x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
#define AbsMask 0x7fffffffffffffff
#define Half 0x3fe0000000000000
#define OFlowBound \
0x40862e42fefa39f0 /* 0x1.62e42fefa39fp+9, above which using expm1 results \
in NaN. */
double
__exp_dd (double, double);
/* Approximation for double-precision sinh(x) using expm1.
sinh(x) = (exp(x) - exp(-x)) / 2.
The greatest observed error is 2.57 ULP:
__v_sinh(0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2
want 0x1.ab34e59d678d9p-2. */
double
sinh (double x)
{
uint64_t ix = asuint64 (x);
uint64_t iax = ix & AbsMask;
double ax = asdouble (iax);
uint64_t sign = ix & ~AbsMask;
double halfsign = asdouble (Half | sign);
if (unlikely (iax >= OFlowBound))
{
/* Special values and overflow. */
if (unlikely (iax > 0x7ff0000000000000))
return __math_invalidf (x);
/* expm1 overflows a little before sinh. We have to fill this
gap by using a different algorithm, in this case we use a
double-precision exp helper. For large x sinh(x) is dominated
by exp(x), however we cannot compute exp without overflow
either. We use the identity: exp(a) = (exp(a / 2)) ^ 2
to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0
~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. */
double e = __exp_dd (ax / 2, 0);
return (e * halfsign) * e;
}
/* Use expm1f to retain acceptable precision for small numbers.
Let t = e^(|x|) - 1. */
double t = expm1 (ax);
/* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0
(t + t / (t + 1)) / -2 for x < 0. */
return (t + t / (t + 1)) * halfsign;
}
PL_SIG (S, D, 1, sinh, -10.0, 10.0)
PL_TEST_ULP (sinh, 2.08)
PL_TEST_INTERVAL (sinh, 0, 0x1p-51, 100)
PL_TEST_INTERVAL (sinh, -0, -0x1p-51, 100)
PL_TEST_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000)
PL_TEST_INTERVAL (sinh, -0x1p-51, -0x1.62e42fefa39fp+9, 100000)
PL_TEST_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000)
PL_TEST_INTERVAL (sinh, -0x1.62e42fefa39fp+9, -inf, 1000)
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