1 /* 2 * Double-precision asinh(x) function 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 #include "poly_scalar_f64.h" 8 #include "math_config.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 #define AbsMask 0x7fffffffffffffff 13 #define ExpM26 0x3e50000000000000 /* asuint64(0x1.0p-26). */ 14 #define One 0x3ff0000000000000 /* asuint64(1.0). */ 15 #define Exp511 0x5fe0000000000000 /* asuint64(0x1.0p511). */ 16 #define Ln2 0x1.62e42fefa39efp-1 17 18 double 19 optr_aor_log_f64 (double); 20 21 /* Scalar double-precision asinh implementation. This routine uses different 22 approaches on different intervals: 23 24 |x| < 2^-26: Return x. Function is exact in this region. 25 26 |x| < 1: Use custom order-17 polynomial. This is least accurate close to 1. 27 The largest observed error in this region is 1.47 ULPs: 28 asinh(0x1.fdfcd00cc1e6ap-1) got 0x1.c1d6bf874019bp-1 29 want 0x1.c1d6bf874019cp-1. 30 31 |x| < 2^511: Upper bound of this region is close to sqrt(DBL_MAX). Calculate 32 the result directly using the definition asinh(x) = ln(x + sqrt(x*x + 1)). 33 The largest observed error in this region is 2.03 ULPs: 34 asinh(-0x1.00094e0f39574p+0) got -0x1.c3508eb6a681ep-1 35 want -0x1.c3508eb6a682p-1. 36 37 |x| >= 2^511: We cannot square x without overflow at a low 38 cost. At very large x, asinh(x) ~= ln(2x). At huge x we cannot 39 even double x without overflow, so calculate this as ln(x) + 40 ln(2). The largest observed error in this region is 0.98 ULPs at many 41 values, for instance: 42 asinh(0x1.5255a4cf10319p+975) got 0x1.52652f4cb26cbp+9 43 want 0x1.52652f4cb26ccp+9. */ 44 double 45 asinh (double x) 46 { 47 uint64_t ix = asuint64 (x); 48 uint64_t ia = ix & AbsMask; 49 double ax = asdouble (ia); 50 uint64_t sign = ix & ~AbsMask; 51 52 if (ia < ExpM26) 53 { 54 return x; 55 } 56 57 if (ia < One) 58 { 59 double x2 = x * x; 60 double z2 = x2 * x2; 61 double z4 = z2 * z2; 62 double z8 = z4 * z4; 63 double p = estrin_17_f64 (x2, z2, z4, z8, z8 * z8, __asinh_data.poly); 64 double y = fma (p, x2 * ax, ax); 65 return asdouble (asuint64 (y) | sign); 66 } 67 68 if (unlikely (ia >= Exp511)) 69 { 70 return asdouble (asuint64 (optr_aor_log_f64 (ax) + Ln2) | sign); 71 } 72 73 return asdouble (asuint64 (optr_aor_log_f64 (ax + sqrt (ax * ax + 1))) 74 | sign); 75 } 76 77 PL_SIG (S, D, 1, asinh, -10.0, 10.0) 78 PL_TEST_ULP (asinh, 1.54) 79 PL_TEST_INTERVAL (asinh, -0x1p-26, 0x1p-26, 50000) 80 PL_TEST_INTERVAL (asinh, 0x1p-26, 1.0, 40000) 81 PL_TEST_INTERVAL (asinh, -0x1p-26, -1.0, 10000) 82 PL_TEST_INTERVAL (asinh, 1.0, 100.0, 40000) 83 PL_TEST_INTERVAL (asinh, -1.0, -100.0, 10000) 84 PL_TEST_INTERVAL (asinh, 100.0, inf, 50000) 85 PL_TEST_INTERVAL (asinh, -100.0, -inf, 10000) 86