1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision tanh(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li #include "math_config.h"
8*412f47f9SXin Li #include "poly_scalar_f64.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define AbsMask 0x7fffffffffffffff
13*412f47f9SXin Li #define InvLn2 0x1.71547652b82fep0
14*412f47f9SXin Li #define Ln2hi 0x1.62e42fefa39efp-1
15*412f47f9SXin Li #define Ln2lo 0x1.abc9e3b39803fp-56
16*412f47f9SXin Li #define Shift 0x1.8p52
17*412f47f9SXin Li
18*412f47f9SXin Li #define BoringBound 0x403241bf835f9d5f /* asuint64 (0x1.241bf835f9d5fp+4). */
19*412f47f9SXin Li #define TinyBound 0x3e40000000000000 /* asuint64 (0x1p-27). */
20*412f47f9SXin Li #define One 0x3ff0000000000000
21*412f47f9SXin Li
22*412f47f9SXin Li static inline double
expm1_inline(double x)23*412f47f9SXin Li expm1_inline (double x)
24*412f47f9SXin Li {
25*412f47f9SXin Li /* Helper routine for calculating exp(x) - 1. Copied from expm1_2u5.c, with
26*412f47f9SXin Li several simplifications:
27*412f47f9SXin Li - No special-case handling for tiny or special values.
28*412f47f9SXin Li - Simpler combination of p and t in final stage of the algorithm.
29*412f47f9SXin Li - Use shift-and-add instead of ldexp to calculate t. */
30*412f47f9SXin Li
31*412f47f9SXin Li /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */
32*412f47f9SXin Li double j = fma (InvLn2, x, Shift) - Shift;
33*412f47f9SXin Li int64_t i = j;
34*412f47f9SXin Li double f = fma (j, -Ln2hi, x);
35*412f47f9SXin Li f = fma (j, -Ln2lo, f);
36*412f47f9SXin Li
37*412f47f9SXin Li /* Approximate expm1(f) using polynomial. */
38*412f47f9SXin Li double f2 = f * f;
39*412f47f9SXin Li double f4 = f2 * f2;
40*412f47f9SXin Li double p = fma (f2, estrin_10_f64 (f, f2, f4, f4 * f4, __expm1_poly), f);
41*412f47f9SXin Li
42*412f47f9SXin Li /* t = 2 ^ i. */
43*412f47f9SXin Li double t = asdouble ((uint64_t) (i + 1023) << 52);
44*412f47f9SXin Li /* expm1(x) = p * t + (t - 1). */
45*412f47f9SXin Li return fma (p, t, t - 1);
46*412f47f9SXin Li }
47*412f47f9SXin Li
48*412f47f9SXin Li /* Approximation for double-precision tanh(x), using a simplified version of
49*412f47f9SXin Li expm1. The greatest observed error is 2.77 ULP:
50*412f47f9SXin Li tanh(-0x1.c4a4ca0f9f3b7p-3) got -0x1.bd6a21a163627p-3
51*412f47f9SXin Li want -0x1.bd6a21a163624p-3. */
52*412f47f9SXin Li double
tanh(double x)53*412f47f9SXin Li tanh (double x)
54*412f47f9SXin Li {
55*412f47f9SXin Li uint64_t ix = asuint64 (x);
56*412f47f9SXin Li uint64_t ia = ix & AbsMask;
57*412f47f9SXin Li uint64_t sign = ix & ~AbsMask;
58*412f47f9SXin Li
59*412f47f9SXin Li if (unlikely (ia > BoringBound))
60*412f47f9SXin Li {
61*412f47f9SXin Li if (ia > 0x7ff0000000000000)
62*412f47f9SXin Li return __math_invalid (x);
63*412f47f9SXin Li return asdouble (One | sign);
64*412f47f9SXin Li }
65*412f47f9SXin Li
66*412f47f9SXin Li if (unlikely (ia < TinyBound))
67*412f47f9SXin Li return x;
68*412f47f9SXin Li
69*412f47f9SXin Li /* tanh(x) = (e^2x - 1) / (e^2x + 1). */
70*412f47f9SXin Li double q = expm1_inline (2 * x);
71*412f47f9SXin Li return q / (q + 2);
72*412f47f9SXin Li }
73*412f47f9SXin Li
74*412f47f9SXin Li PL_SIG (S, D, 1, tanh, -10.0, 10.0)
75*412f47f9SXin Li PL_TEST_ULP (tanh, 2.27)
76*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanh, 0, TinyBound, 1000)
77*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanh, TinyBound, BoringBound, 100000)
78*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanh, BoringBound, inf, 1000)
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