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 "estrin.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 #define C(i) __expm1_poly[i] 18 19 #define BoringBound 0x403241bf835f9d5f /* asuint64 (0x1.241bf835f9d5fp+4). */ 20 #define TinyBound 0x3e40000000000000 /* asuint64 (0x1p-27). */ 21 #define One 0x3ff0000000000000 22 23 static inline double 24 expm1_inline (double x) 25 { 26 /* Helper routine for calculating exp(x) - 1. Copied from expm1_2u5.c, with 27 several simplifications: 28 - No special-case handling for tiny or special values. 29 - Simpler combination of p and t in final stage of the algorithm. 30 - Use shift-and-add instead of ldexp to calculate t. */ 31 32 /* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */ 33 double j = fma (InvLn2, x, Shift) - Shift; 34 int64_t i = j; 35 double f = fma (j, -Ln2hi, x); 36 f = fma (j, -Ln2lo, f); 37 38 /* Approximate expm1(f) using polynomial. */ 39 double f2 = f * f; 40 double f4 = f2 * f2; 41 double p = fma (f2, ESTRIN_10 (f, f2, f4, f4 * f4, C), f); 42 43 /* t = 2 ^ i. */ 44 double t = asdouble ((uint64_t) (i + 1023) << 52); 45 /* expm1(x) = p * t + (t - 1). */ 46 return fma (p, t, t - 1); 47 } 48 49 /* Approximation for double-precision tanh(x), using a simplified version of 50 expm1. The greatest observed error is 2.75 ULP: 51 tanh(-0x1.c143c3a44e087p-3) got -0x1.ba31ba4691ab7p-3 52 want -0x1.ba31ba4691ab4p-3. */ 53 double 54 tanh (double x) 55 { 56 uint64_t ix = asuint64 (x); 57 uint64_t ia = ix & AbsMask; 58 uint64_t sign = ix & ~AbsMask; 59 60 if (unlikely (ia > BoringBound)) 61 { 62 if (ia > 0x7ff0000000000000) 63 return __math_invalid (x); 64 return asdouble (One | sign); 65 } 66 67 if (unlikely (ia < TinyBound)) 68 return x; 69 70 /* tanh(x) = (e^2x - 1) / (e^2x + 1). */ 71 double q = expm1_inline (2 * x); 72 return q / (q + 2); 73 } 74 75 PL_SIG (S, D, 1, tanh, -10.0, 10.0) 76 PL_TEST_ULP (tanh, 2.26) 77 PL_TEST_INTERVAL (tanh, 0, TinyBound, 1000) 78 PL_TEST_INTERVAL (tanh, -0, -TinyBound, 1000) 79 PL_TEST_INTERVAL (tanh, TinyBound, BoringBound, 100000) 80 PL_TEST_INTERVAL (tanh, -TinyBound, -BoringBound, 100000) 81 PL_TEST_INTERVAL (tanh, BoringBound, inf, 1000) 82 PL_TEST_INTERVAL (tanh, -BoringBound, -inf, 1000) 83