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/*
* Double-precision asinh(x) function
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "estrin.h"
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
#define AbsMask 0x7fffffffffffffff
#define ExpM26 0x3e50000000000000 /* asuint64(0x1.0p-26). */
#define One 0x3ff0000000000000 /* asuint64(1.0). */
#define Exp511 0x5fe0000000000000 /* asuint64(0x1.0p511). */
#define Ln2 0x1.62e42fefa39efp-1
double
optr_aor_log_f64 (double);
/* Scalar double-precision asinh implementation. This routine uses different
approaches on different intervals:
|x| < 2^-26: Return x. Function is exact in this region.
|x| < 1: Use custom order-17 polynomial. This is least accurate close to 1.
The largest observed error in this region is 1.47 ULPs:
asinh(0x1.fdfcd00cc1e6ap-1) got 0x1.c1d6bf874019bp-1
want 0x1.c1d6bf874019cp-1.
|x| < 2^511: Upper bound of this region is close to sqrt(DBL_MAX). Calculate
the result directly using the definition asinh(x) = ln(x + sqrt(x*x + 1)).
The largest observed error in this region is 2.03 ULPs:
asinh(-0x1.00094e0f39574p+0) got -0x1.c3508eb6a681ep-1
want -0x1.c3508eb6a682p-1.
|x| >= 2^511: We cannot square x without overflow at a low
cost. At very large x, asinh(x) ~= ln(2x). At huge x we cannot
even double x without overflow, so calculate this as ln(x) +
ln(2). The largest observed error in this region is 0.98 ULPs at many
values, for instance:
asinh(0x1.5255a4cf10319p+975) got 0x1.52652f4cb26cbp+9
want 0x1.52652f4cb26ccp+9. */
double
asinh (double x)
{
uint64_t ix = asuint64 (x);
uint64_t ia = ix & AbsMask;
double ax = asdouble (ia);
uint64_t sign = ix & ~AbsMask;
if (ia < ExpM26)
{
return x;
}
if (ia < One)
{
double x2 = x * x;
double z2 = x2 * x2;
double z4 = z2 * z2;
double z8 = z4 * z4;
#define C(i) __asinh_data.poly[i]
double p = ESTRIN_17 (x2, z2, z4, z8, z8 * z8, C);
double y = fma (p, x2 * ax, ax);
return asdouble (asuint64 (y) | sign);
}
if (unlikely (ia >= Exp511))
{
return asdouble (asuint64 (optr_aor_log_f64 (ax) + Ln2) | sign);
}
return asdouble (asuint64 (optr_aor_log_f64 (ax + sqrt (ax * ax + 1)))
| sign);
}
PL_SIG (S, D, 1, asinh, -10.0, 10.0)
PL_TEST_ULP (asinh, 1.54)
PL_TEST_INTERVAL (asinh, -0x1p-26, 0x1p-26, 50000)
PL_TEST_INTERVAL (asinh, 0x1p-26, 1.0, 40000)
PL_TEST_INTERVAL (asinh, -0x1p-26, -1.0, 10000)
PL_TEST_INTERVAL (asinh, 1.0, 100.0, 40000)
PL_TEST_INTERVAL (asinh, -1.0, -100.0, 10000)
PL_TEST_INTERVAL (asinh, 100.0, inf, 50000)
PL_TEST_INTERVAL (asinh, -100.0, -inf, 10000)
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