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