1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-precision acosh(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2022-2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "math_config.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define Ln2 (0x1.62e4p-1f)
13*412f47f9SXin Li #define MinusZero 0x80000000
14*412f47f9SXin Li #define SquareLim 0x5f800000 /* asuint(0x1p64). */
15*412f47f9SXin Li #define Two 0x40000000
16*412f47f9SXin Li
17*412f47f9SXin Li /* Single-precision log from math/. */
18*412f47f9SXin Li float
19*412f47f9SXin Li optr_aor_log_f32 (float);
20*412f47f9SXin Li
21*412f47f9SXin Li /* Single-precision log(1+x) from pl/math. */
22*412f47f9SXin Li float
23*412f47f9SXin Li log1pf (float);
24*412f47f9SXin Li
25*412f47f9SXin Li /* acoshf approximation using a variety of approaches on different intervals:
26*412f47f9SXin Li
27*412f47f9SXin Li x >= 2^64: We cannot square x without overflow. For huge x, sqrt(x*x - 1) is
28*412f47f9SXin Li close enough to x that we can calculate the result by ln(2x) == ln(x) +
29*412f47f9SXin Li ln(2). The greatest error in the region is 0.94 ULP:
30*412f47f9SXin Li acoshf(0x1.15f706p+92) got 0x1.022e14p+6 want 0x1.022e16p+6.
31*412f47f9SXin Li
32*412f47f9SXin Li x > 2: Calculate the result directly using definition of asinh(x) = ln(x +
33*412f47f9SXin Li sqrt(x*x - 1)). Greatest error in this region is 1.30 ULP:
34*412f47f9SXin Li acoshf(0x1.249d8p+1) got 0x1.77e1aep+0 want 0x1.77e1bp+0.
35*412f47f9SXin Li
36*412f47f9SXin Li 0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is
37*412f47f9SXin Li undefined. For 1 <= x <= 2, the greatest error is 2.78 ULP:
38*412f47f9SXin Li acoshf(0x1.07887p+0) got 0x1.ef9e9cp-3 want 0x1.ef9ea2p-3. */
39*412f47f9SXin Li float
acoshf(float x)40*412f47f9SXin Li acoshf (float x)
41*412f47f9SXin Li {
42*412f47f9SXin Li uint32_t ix = asuint (x);
43*412f47f9SXin Li
44*412f47f9SXin Li if (unlikely (ix >= MinusZero))
45*412f47f9SXin Li return __math_invalidf (x);
46*412f47f9SXin Li
47*412f47f9SXin Li if (unlikely (ix >= SquareLim))
48*412f47f9SXin Li return optr_aor_log_f32 (x) + Ln2;
49*412f47f9SXin Li
50*412f47f9SXin Li if (ix > Two)
51*412f47f9SXin Li return optr_aor_log_f32 (x + sqrtf (x * x - 1));
52*412f47f9SXin Li
53*412f47f9SXin Li float xm1 = x - 1;
54*412f47f9SXin Li return log1pf (xm1 + sqrtf (2 * xm1 + xm1 * xm1));
55*412f47f9SXin Li }
56*412f47f9SXin Li
57*412f47f9SXin Li PL_SIG (S, F, 1, acosh, 1.0, 10.0)
58*412f47f9SXin Li PL_TEST_ULP (acoshf, 2.30)
59*412f47f9SXin Li PL_TEST_INTERVAL (acoshf, 0, 1, 100)
60*412f47f9SXin Li PL_TEST_INTERVAL (acoshf, 1, 2, 10000)
61*412f47f9SXin Li PL_TEST_INTERVAL (acoshf, 2, 0x1p64, 100000)
62*412f47f9SXin Li PL_TEST_INTERVAL (acoshf, 0x1p64, inf, 100000)
63*412f47f9SXin Li PL_TEST_INTERVAL (acoshf, -0, -inf, 10000)
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