1 /* 2 * Double-precision asinh(x) function 3 * 4 * Copyright (c) 2022-2024, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 #include "mathlib.h" 8 #include "poly_scalar_f64.h" 9 #include "math_config.h" 10 #include "test_sig.h" 11 #include "test_defs.h" 12 13 #define AbsMask 0x7fffffffffffffff 14 #define ExpM26 0x3e50000000000000 /* asuint64(0x1.0p-26). */ 15 #define One 0x3ff0000000000000 /* asuint64(1.0). */ 16 #define Exp511 0x5fe0000000000000 /* asuint64(0x1.0p511). */ 17 #define Ln2 0x1.62e42fefa39efp-1 18 19 /* Scalar double-precision asinh implementation. This routine uses different 20 approaches on different intervals: 21 22 |x| < 2^-26: Return x. Function is exact in this region. 23 24 |x| < 1: Use custom order-17 polynomial. This is least accurate close to 1. 25 The largest observed error in this region is 1.47 ULPs: 26 asinh(0x1.fdfcd00cc1e6ap-1) got 0x1.c1d6bf874019bp-1 27 want 0x1.c1d6bf874019cp-1. 28 29 |x| < 2^511: Upper bound of this region is close to sqrt(DBL_MAX). Calculate 30 the result directly using the definition asinh(x) = ln(x + sqrt(x*x + 1)). 31 The largest observed error in this region is 2.03 ULPs: 32 asinh(-0x1.00094e0f39574p+0) got -0x1.c3508eb6a681ep-1 33 want -0x1.c3508eb6a682p-1. 34 35 |x| >= 2^511: We cannot square x without overflow at a low 36 cost. At very large x, asinh(x) ~= ln(2x). At huge x we cannot 37 even double x without overflow, so calculate this as ln(x) + 38 ln(2). The largest observed error in this region is 0.98 ULPs at many 39 values, for instance: 40 asinh(0x1.5255a4cf10319p+975) got 0x1.52652f4cb26cbp+9 41 want 0x1.52652f4cb26ccp+9. */ 42 double 43 asinh (double x) 44 { 45 uint64_t ix = asuint64 (x); 46 uint64_t ia = ix & AbsMask; 47 double ax = asdouble (ia); 48 uint64_t sign = ix & ~AbsMask; 49 50 if (ia < ExpM26) 51 { 52 return x; 53 } 54 55 if (ia < One) 56 { 57 double x2 = x * x; 58 double z2 = x2 * x2; 59 double z4 = z2 * z2; 60 double z8 = z4 * z4; 61 double p = estrin_17_f64 (x2, z2, z4, z8, z8 * z8, __asinh_data.poly); 62 double y = fma (p, x2 * ax, ax); 63 return asdouble (asuint64 (y) | sign); 64 } 65 66 if (unlikely (ia >= Exp511)) 67 { 68 return asdouble (asuint64 (log (ax) + Ln2) | sign); 69 } 70 71 return asdouble (asuint64 (log (ax + sqrt (ax * ax + 1))) | sign); 72 } 73 74 TEST_SIG (S, D, 1, asinh, -10.0, 10.0) 75 TEST_ULP (asinh, 1.54) 76 TEST_INTERVAL (asinh, -0x1p-26, 0x1p-26, 50000) 77 TEST_INTERVAL (asinh, 0x1p-26, 1.0, 40000) 78 TEST_INTERVAL (asinh, -0x1p-26, -1.0, 10000) 79 TEST_INTERVAL (asinh, 1.0, 100.0, 40000) 80 TEST_INTERVAL (asinh, -1.0, -100.0, 10000) 81 TEST_INTERVAL (asinh, 100.0, inf, 50000) 82 TEST_INTERVAL (asinh, -100.0, -inf, 10000) 83