1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-precision tanh(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner #include "math_config.h"
8*f3087befSAndrew Turner #include "poly_scalar_f64.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner
12*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff
13*f3087befSAndrew Turner #define InvLn2 0x1.71547652b82fep0
14*f3087befSAndrew Turner #define Ln2hi 0x1.62e42fefa39efp-1
15*f3087befSAndrew Turner #define Ln2lo 0x1.abc9e3b39803fp-56
16*f3087befSAndrew Turner #define Shift 0x1.8p52
17*f3087befSAndrew Turner
18*f3087befSAndrew Turner /* asuint64 (0x1.241bf835f9d5fp+4). */
19*f3087befSAndrew Turner #define BoringBound 0x403241bf835f9d5f
20*f3087befSAndrew Turner /* asuint64 (0x1p-27). */
21*f3087befSAndrew Turner #define TinyBound 0x3e40000000000000
22*f3087befSAndrew Turner #define One 0x3ff0000000000000
23*f3087befSAndrew Turner
24*f3087befSAndrew Turner static inline double
expm1_inline(double x)25*f3087befSAndrew Turner expm1_inline (double x)
26*f3087befSAndrew Turner {
27*f3087befSAndrew Turner /* Helper routine for calculating exp(x) - 1. Copied from expm1_2u5.c, with
28*f3087befSAndrew Turner several simplifications:
29*f3087befSAndrew Turner - No special-case handling for tiny or special values.
30*f3087befSAndrew Turner - Simpler combination of p and t in final stage of the algorithm.
31*f3087befSAndrew Turner - Use shift-and-add instead of ldexp to calculate t. */
32*f3087befSAndrew Turner
33*f3087befSAndrew Turner /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */
34*f3087befSAndrew Turner double j = fma (InvLn2, x, Shift) - Shift;
35*f3087befSAndrew Turner int64_t i = j;
36*f3087befSAndrew Turner double f = fma (j, -Ln2hi, x);
37*f3087befSAndrew Turner f = fma (j, -Ln2lo, f);
38*f3087befSAndrew Turner
39*f3087befSAndrew Turner /* Approximate expm1(f) using polynomial. */
40*f3087befSAndrew Turner double f2 = f * f;
41*f3087befSAndrew Turner double f4 = f2 * f2;
42*f3087befSAndrew Turner double p = fma (f2, estrin_10_f64 (f, f2, f4, f4 * f4, __expm1_poly), f);
43*f3087befSAndrew Turner
44*f3087befSAndrew Turner /* t = 2 ^ i. */
45*f3087befSAndrew Turner double t = asdouble ((uint64_t) (i + 1023) << 52);
46*f3087befSAndrew Turner /* expm1(x) = p * t + (t - 1). */
47*f3087befSAndrew Turner return fma (p, t, t - 1);
48*f3087befSAndrew Turner }
49*f3087befSAndrew Turner
50*f3087befSAndrew Turner /* Approximation for double-precision tanh(x), using a simplified version of
51*f3087befSAndrew Turner expm1. The greatest observed error is 2.77 ULP:
52*f3087befSAndrew Turner tanh(-0x1.c4a4ca0f9f3b7p-3) got -0x1.bd6a21a163627p-3
53*f3087befSAndrew Turner want -0x1.bd6a21a163624p-3. */
54*f3087befSAndrew Turner double
tanh(double x)55*f3087befSAndrew Turner tanh (double x)
56*f3087befSAndrew Turner {
57*f3087befSAndrew Turner uint64_t ix = asuint64 (x);
58*f3087befSAndrew Turner uint64_t ia = ix & AbsMask;
59*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask;
60*f3087befSAndrew Turner
61*f3087befSAndrew Turner if (unlikely (ia > BoringBound))
62*f3087befSAndrew Turner {
63*f3087befSAndrew Turner if (ia > 0x7ff0000000000000)
64*f3087befSAndrew Turner return __math_invalid (x);
65*f3087befSAndrew Turner return asdouble (One | sign);
66*f3087befSAndrew Turner }
67*f3087befSAndrew Turner
68*f3087befSAndrew Turner if (unlikely (ia < TinyBound))
69*f3087befSAndrew Turner return x;
70*f3087befSAndrew Turner
71*f3087befSAndrew Turner /* tanh(x) = (e^2x - 1) / (e^2x + 1). */
72*f3087befSAndrew Turner double q = expm1_inline (2 * x);
73*f3087befSAndrew Turner return q / (q + 2);
74*f3087befSAndrew Turner }
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner TEST_SIG (S, D, 1, tanh, -10.0, 10.0)
77*f3087befSAndrew Turner TEST_ULP (tanh, 2.27)
78*f3087befSAndrew Turner TEST_SYM_INTERVAL (tanh, 0, TinyBound, 1000)
79*f3087befSAndrew Turner TEST_SYM_INTERVAL (tanh, TinyBound, BoringBound, 100000)
80*f3087befSAndrew Turner TEST_SYM_INTERVAL (tanh, BoringBound, inf, 1000)
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