1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-precision tanh(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 #include "math_config.h"
8*412f47f9SXin Li #include "pl_sig.h"
9*412f47f9SXin Li #include "pl_test.h"
10*412f47f9SXin Li
11*412f47f9SXin Li #define BoringBound \
12*412f47f9SXin Li 0x41102cb3 /* 0x1.205966p+3, above which tanhf rounds to 1 (or -1 for \
13*412f47f9SXin Li negative). */
14*412f47f9SXin Li #define AbsMask 0x7fffffff
15*412f47f9SXin Li #define One 0x3f800000
16*412f47f9SXin Li
17*412f47f9SXin Li #define Shift (0x1.8p23f)
18*412f47f9SXin Li #define InvLn2 (0x1.715476p+0f)
19*412f47f9SXin Li #define Ln2hi (0x1.62e4p-1f)
20*412f47f9SXin Li #define Ln2lo (0x1.7f7d1cp-20f)
21*412f47f9SXin Li
22*412f47f9SXin Li #define C(i) __expm1f_poly[i]
23*412f47f9SXin Li
24*412f47f9SXin Li static inline float
expm1f_inline(float x)25*412f47f9SXin Li expm1f_inline (float x)
26*412f47f9SXin Li {
27*412f47f9SXin Li /* Helper routine for calculating exp(x) - 1.
28*412f47f9SXin Li Copied from expm1f_1u6.c, with several simplifications:
29*412f47f9SXin Li - No special-case handling for tiny or special values, instead return early
30*412f47f9SXin Li from the main routine.
31*412f47f9SXin Li - No special handling for large values:
32*412f47f9SXin Li - No early return for infinity.
33*412f47f9SXin Li - Simpler combination of p and t in final stage of algorithm.
34*412f47f9SXin Li - |i| < 27, so can calculate t by simpler shift-and-add, instead of
35*412f47f9SXin Li ldexpf (same as vector algorithm). */
36*412f47f9SXin Li
37*412f47f9SXin Li /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */
38*412f47f9SXin Li float j = fmaf (InvLn2, x, Shift) - Shift;
39*412f47f9SXin Li int32_t i = j;
40*412f47f9SXin Li float f = fmaf (j, -Ln2hi, x);
41*412f47f9SXin Li f = fmaf (j, -Ln2lo, f);
42*412f47f9SXin Li
43*412f47f9SXin Li /* Approximate expm1(f) with polynomial P, expm1(f) ~= f + f^2 * P(f).
44*412f47f9SXin Li Uses Estrin scheme, where the main expm1f routine uses Horner. */
45*412f47f9SXin Li float f2 = f * f;
46*412f47f9SXin Li float p_01 = fmaf (f, C (1), C (0));
47*412f47f9SXin Li float p_23 = fmaf (f, C (3), C (2));
48*412f47f9SXin Li float p = fmaf (f2, p_23, p_01);
49*412f47f9SXin Li p = fmaf (f2 * f2, C (4), p);
50*412f47f9SXin Li p = fmaf (f2, p, f);
51*412f47f9SXin Li
52*412f47f9SXin Li /* t = 2^i. */
53*412f47f9SXin Li float t = asfloat ((uint32_t) (i + 127) << 23);
54*412f47f9SXin Li /* expm1(x) ~= p * t + (t - 1). */
55*412f47f9SXin Li return fmaf (p, t, t - 1);
56*412f47f9SXin Li }
57*412f47f9SXin Li
58*412f47f9SXin Li /* Approximation for single-precision tanh(x), using a simplified version of
59*412f47f9SXin Li expm1f. The maximum error is 2.58 ULP:
60*412f47f9SXin Li tanhf(0x1.fa5eep-5) got 0x1.f9ba02p-5
61*412f47f9SXin Li want 0x1.f9ba08p-5. */
62*412f47f9SXin Li float
tanhf(float x)63*412f47f9SXin Li tanhf (float x)
64*412f47f9SXin Li {
65*412f47f9SXin Li uint32_t ix = asuint (x);
66*412f47f9SXin Li uint32_t iax = ix & AbsMask;
67*412f47f9SXin Li uint32_t sign = ix & ~AbsMask;
68*412f47f9SXin Li
69*412f47f9SXin Li if (unlikely (iax > BoringBound))
70*412f47f9SXin Li {
71*412f47f9SXin Li if (iax > 0x7f800000)
72*412f47f9SXin Li return __math_invalidf (x);
73*412f47f9SXin Li return asfloat (One | sign);
74*412f47f9SXin Li }
75*412f47f9SXin Li
76*412f47f9SXin Li if (unlikely (iax < 0x34000000))
77*412f47f9SXin Li return x;
78*412f47f9SXin Li
79*412f47f9SXin Li /* tanh(x) = (e^2x - 1) / (e^2x + 1). */
80*412f47f9SXin Li float q = expm1f_inline (2 * x);
81*412f47f9SXin Li return q / (q + 2);
82*412f47f9SXin Li }
83*412f47f9SXin Li
84*412f47f9SXin Li PL_SIG (S, F, 1, tanh, -10.0, 10.0)
85*412f47f9SXin Li PL_TEST_ULP (tanhf, 2.09)
86*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanhf, 0, 0x1p-23, 1000)
87*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanhf, 0x1p-23, 0x1.205966p+3, 100000)
88*412f47f9SXin Li PL_TEST_SYM_INTERVAL (tanhf, 0x1.205966p+3, inf, 100)
89