pl/math/acosh_3u.c

/*
 * Double-precision acosh(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 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));
}

PL_SIG (S, D, 1, acosh, 1.0, 10.0)
PL_TEST_ULP (acosh, 2.19)
PL_TEST_INTERVAL (acosh, 0, 1, 10000)
PL_TEST_INTERVAL (acosh, 1, 2, 100000)
PL_TEST_INTERVAL (acosh, 2, 0x1p511, 100000)
PL_TEST_INTERVAL (acosh, 0x1p511, inf, 100000)
PL_TEST_INTERVAL (acosh, -0, -inf, 10000)