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/*
* Double-precision acosh(x) function.
* Copyright (c) 2022, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#define Ln2 (0x1.62e42fefa39efp-1)
#define MinusZero (0x8000000000000000)
#define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511). */
#define Two (0x4000000000000000) /* asuint64(2.0). */
double
optr_aor_log_f64 (double);
double
log1p (double);
/* acosh approximation using a variety of approaches on different intervals:
acosh(x) = ln(x + sqrt(x * x - 1)).
x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) is
close enough to x that we can calculate the result by ln(2x) == ln(x) +
ln(2). The greatest observed error in this region is 0.98 ULP:
acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9
want 0x1.28066a11a7c8p+9.
x > 2: Calculate the result directly using definition of acosh(x). Greatest
observed error in this region is 1.33 ULP:
acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0
want 0x1.71a06f50c34b6p+0.
0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is
undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP:
acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3
want 0x1.e4d9bd20684f6p-3. */
double
acosh (double x)
{
uint64_t ix = asuint64 (x);
if (unlikely (ix >= MinusZero))
return __math_invalid (x);
if (unlikely (ix >= SquareLim))
return optr_aor_log_f64 (x) + Ln2;
if (ix >= Two)
return optr_aor_log_f64 (x + sqrt (x * x - 1));
double xm1 = x - 1;
return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1));
}
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