1 /* 2 * Double-precision sinh(x) function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "math_config.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 #define AbsMask 0x7fffffffffffffff 13 #define Half 0x3fe0000000000000 14 #define OFlowBound \ 15 0x40862e42fefa39f0 /* 0x1.62e42fefa39fp+9, above which using expm1 results \ 16 in NaN. */ 17 18 double 19 __exp_dd (double, double); 20 21 /* Approximation for double-precision sinh(x) using expm1. 22 sinh(x) = (exp(x) - exp(-x)) / 2. 23 The greatest observed error is 2.57 ULP: 24 __v_sinh(0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2 25 want 0x1.ab34e59d678d9p-2. */ 26 double 27 sinh (double x) 28 { 29 uint64_t ix = asuint64 (x); 30 uint64_t iax = ix & AbsMask; 31 double ax = asdouble (iax); 32 uint64_t sign = ix & ~AbsMask; 33 double halfsign = asdouble (Half | sign); 34 35 if (unlikely (iax >= OFlowBound)) 36 { 37 /* Special values and overflow. */ 38 if (unlikely (iax > 0x7ff0000000000000)) 39 return __math_invalidf (x); 40 /* expm1 overflows a little before sinh. We have to fill this 41 gap by using a different algorithm, in this case we use a 42 double-precision exp helper. For large x sinh(x) is dominated 43 by exp(x), however we cannot compute exp without overflow 44 either. We use the identity: exp(a) = (exp(a / 2)) ^ 2 45 to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0 46 ~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. */ 47 double e = __exp_dd (ax / 2, 0); 48 return (e * halfsign) * e; 49 } 50 51 /* Use expm1f to retain acceptable precision for small numbers. 52 Let t = e^(|x|) - 1. */ 53 double t = expm1 (ax); 54 /* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0 55 (t + t / (t + 1)) / -2 for x < 0. */ 56 return (t + t / (t + 1)) * halfsign; 57 } 58 59 PL_SIG (S, D, 1, sinh, -10.0, 10.0) 60 PL_TEST_ULP (sinh, 2.08) 61 PL_TEST_INTERVAL (sinh, 0, 0x1p-51, 100) 62 PL_TEST_INTERVAL (sinh, -0, -0x1p-51, 100) 63 PL_TEST_INTERVAL (sinh, 0x1p-51, 0x1.62e42fefa39fp+9, 100000) 64 PL_TEST_INTERVAL (sinh, -0x1p-51, -0x1.62e42fefa39fp+9, 100000) 65 PL_TEST_INTERVAL (sinh, 0x1.62e42fefa39fp+9, inf, 1000) 66 PL_TEST_INTERVAL (sinh, -0x1.62e42fefa39fp+9, -inf, 1000) 67