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 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 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) 81