1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision sinh(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "math_config.h" 9*f3087befSAndrew Turner #include "test_sig.h" 10*f3087befSAndrew Turner #include "test_defs.h" 11*f3087befSAndrew Turner #include "exp_inline.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff 14*f3087befSAndrew Turner #define Half 0x3fe0000000000000 15*f3087befSAndrew Turner /* 0x1.62e42fefa39fp+9, above which using expm1 results in NaN. */ 16*f3087befSAndrew Turner #define OFlowBound 0x40862e42fefa39f0 17*f3087befSAndrew Turner 18*f3087befSAndrew Turner /* Approximation for double-precision sinh(x) using expm1. 19*f3087befSAndrew Turner sinh(x) = (exp(x) - exp(-x)) / 2. 20*f3087befSAndrew Turner The greatest observed error is 2.57 ULP: 21*f3087befSAndrew Turner __v_sinh(0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2 22*f3087befSAndrew Turner want 0x1.ab34e59d678d9p-2. */ 23*f3087befSAndrew Turner double 24*f3087befSAndrew Turner sinh (double x) 25*f3087befSAndrew Turner { 26*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 27*f3087befSAndrew Turner uint64_t iax = ix & AbsMask; 28*f3087befSAndrew Turner double ax = asdouble (iax); 29*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask; 30*f3087befSAndrew Turner double halfsign = asdouble (Half | sign); 31*f3087befSAndrew Turner 32*f3087befSAndrew Turner if (unlikely (iax >= OFlowBound)) 33*f3087befSAndrew Turner { 34*f3087befSAndrew Turner /* Special values and overflow. */ 35*f3087befSAndrew Turner if (unlikely (iax > 0x7ff0000000000000)) 36*f3087befSAndrew Turner return __math_invalidf (x); 37*f3087befSAndrew Turner /* expm1 overflows a little before sinh. We have to fill this 38*f3087befSAndrew Turner gap by using a different algorithm, in this case we use a 39*f3087befSAndrew Turner double-precision exp helper. For large x sinh(x) is dominated 40*f3087befSAndrew Turner by exp(x), however we cannot compute exp without overflow 41*f3087befSAndrew Turner either. We use the identity: exp(a) = (exp(a / 2)) ^ 2 42*f3087befSAndrew Turner to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0 43*f3087befSAndrew Turner ~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. */ 44*f3087befSAndrew Turner double e = exp_inline (ax / 2, 0); 45*f3087befSAndrew Turner return (e * halfsign) * e; 46*f3087befSAndrew Turner } 47*f3087befSAndrew Turner 48*f3087befSAndrew Turner /* Use expm1f to retain acceptable precision for small numbers. 49*f3087befSAndrew Turner Let t = e^(|x|) - 1. */ 50*f3087befSAndrew Turner double t = expm1 (ax); 51*f3087befSAndrew Turner /* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0 52*f3087befSAndrew Turner (t + t / (t + 1)) / -2 for x < 0. */ 53*f3087befSAndrew Turner return (t + t / (t + 1)) * halfsign; 54*f3087befSAndrew Turner } 55*f3087befSAndrew Turner 56*f3087befSAndrew Turner TEST_SIG (S, D, 1, sinh, -10.0, 10.0) 57*f3087befSAndrew Turner TEST_ULP (sinh, 2.08) 58*f3087befSAndrew Turner TEST_SYM_INTERVAL (sinh, 0, 0x1p-51, 100) 59*f3087befSAndrew Turner TEST_SYM_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000) 60*f3087befSAndrew Turner TEST_SYM_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000) 61