1 /* 2 * Single-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 0x7fffffff 13 #define Half 0x3f000000 14 #define Expm1OFlowLimit \ 15 0x42b17218 /* 0x1.62e43p+6, 2^7*ln2, minimum value for which expm1f \ 16 overflows. */ 17 #define OFlowLimit \ 18 0x42b2d4fd /* 0x1.65a9fap+6, minimum positive value for which sinhf should \ 19 overflow. */ 20 21 float 22 optr_aor_exp_f32 (float); 23 24 /* Approximation for single-precision sinh(x) using expm1. 25 sinh(x) = (exp(x) - exp(-x)) / 2. 26 The maximum error is 2.26 ULP: 27 sinhf(0x1.e34a9ep-4) got 0x1.e469ep-4 want 0x1.e469e4p-4. */ 28 float 29 sinhf (float x) 30 { 31 uint32_t ix = asuint (x); 32 uint32_t iax = ix & AbsMask; 33 float ax = asfloat (iax); 34 uint32_t sign = ix & ~AbsMask; 35 float halfsign = asfloat (Half | sign); 36 37 if (unlikely (iax >= Expm1OFlowLimit)) 38 { 39 /* Special values and overflow. */ 40 if (iax >= 0x7fc00001 || iax == 0x7f800000) 41 return x; 42 if (iax >= 0x7f800000) 43 return __math_invalidf (x); 44 if (iax >= OFlowLimit) 45 return __math_oflowf (sign); 46 47 /* expm1f overflows a little before sinhf, (~88.7 vs ~89.4). We have to 48 fill this gap by using a different algorithm, in this case we use a 49 double-precision exp helper. For large x sinh(x) dominated by exp(x), 50 however we cannot compute exp without overflow either. We use the 51 identity: 52 exp(a) = (exp(a / 2)) ^ 2. 53 to compute sinh(x) ~= (exp(|x| / 2)) ^ 2 / 2 for x > 0 54 ~= (exp(|x| / 2)) ^ 2 / -2 for x < 0. 55 Greatest error in this region is 1.89 ULP: 56 sinhf(0x1.65898cp+6) got 0x1.f00aep+127 want 0x1.f00adcp+127. */ 57 float e = optr_aor_exp_f32 (ax / 2); 58 return (e * halfsign) * e; 59 } 60 61 /* Use expm1f to retain acceptable precision for small numbers. 62 Let t = e^(|x|) - 1. */ 63 float t = expm1f (ax); 64 /* Then sinh(x) = (t + t / (t + 1)) / 2 for x > 0 65 (t + t / (t + 1)) / -2 for x < 0. */ 66 return (t + t / (t + 1)) * halfsign; 67 } 68 69 PL_SIG (S, F, 1, sinh, -10.0, 10.0) 70 PL_TEST_ULP (sinhf, 1.76) 71 PL_TEST_SYM_INTERVAL (sinhf, 0, 0x1.62e43p+6, 100000) 72 PL_TEST_SYM_INTERVAL (sinhf, 0x1.62e43p+6, 0x1.65a9fap+6, 100) 73 PL_TEST_SYM_INTERVAL (sinhf, 0x1.65a9fap+6, inf, 100) 74