1 /*
2  * Double-precision acosh(x) function.
3  *
4  * Copyright (c) 2022-2023, Arm Limited.
5  * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6  */
7 
8 #include "math_config.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11 
12 #define Ln2 (0x1.62e42fefa39efp-1)
13 #define MinusZero (0x8000000000000000)
14 #define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511).  */
15 #define Two (0x4000000000000000)       /* asuint64(2.0).  */
16 
17 double
18 optr_aor_log_f64 (double);
19 
20 double
21 log1p (double);
22 
23 /* acosh approximation using a variety of approaches on different intervals:
24 
25    acosh(x) = ln(x + sqrt(x * x - 1)).
26 
27    x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) is
28    close enough to x that we can calculate the result by ln(2x) == ln(x) +
29    ln(2). The greatest observed error in this region is 0.98 ULP:
30    acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9
31 				want 0x1.28066a11a7c8p+9.
32 
33    x > 2: Calculate the result directly using definition of acosh(x). Greatest
34    observed error in this region is 1.33 ULP:
35    acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0
36 			      want 0x1.71a06f50c34b6p+0.
37 
38    0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is
39    undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP:
40    acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3
41 			     want 0x1.e4d9bd20684f6p-3.  */
42 double
43 acosh (double x)
44 {
45   uint64_t ix = asuint64 (x);
46 
47   if (unlikely (ix >= MinusZero))
48     return __math_invalid (x);
49 
50   if (unlikely (ix >= SquareLim))
51     return optr_aor_log_f64 (x) + Ln2;
52 
53   if (ix >= Two)
54     return optr_aor_log_f64 (x + sqrt (x * x - 1));
55 
56   double xm1 = x - 1;
57   return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1));
58 }
59 
60 PL_SIG (S, D, 1, acosh, 1.0, 10.0)
61 PL_TEST_ULP (acosh, 2.19)
62 PL_TEST_INTERVAL (acosh, 0, 1, 10000)
63 PL_TEST_INTERVAL (acosh, 1, 2, 100000)
64 PL_TEST_INTERVAL (acosh, 2, 0x1p511, 100000)
65 PL_TEST_INTERVAL (acosh, 0x1p511, inf, 100000)
66 PL_TEST_INTERVAL (acosh, -0, -inf, 10000)
67