1 /* 2 * Double-precision acosh(x) function. 3 * 4 * Copyright (c) 2022-2024, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "mathlib.h" 9 #include "math_config.h" 10 #include "test_sig.h" 11 #include "test_defs.h" 12 13 #define Ln2 (0x1.62e42fefa39efp-1) 14 #define MinusZero (0x8000000000000000) 15 #define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511). */ 16 #define Two (0x4000000000000000) /* asuint64(2.0). */ 17 18 /* acosh approximation using a variety of approaches on different intervals: 19 20 acosh(x) = ln(x + sqrt(x * x - 1)). 21 22 x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) 23 is close enough to x that we can calculate the result by ln(2x) == ln(x) + 24 ln(2). The greatest observed error in this region is 0.98 ULP: 25 acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9 26 want 0x1.28066a11a7c8p+9. 27 28 x > 2: Calculate the result directly using definition of acosh(x). Greatest 29 observed error in this region is 1.33 ULP: 30 acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0 31 want 0x1.71a06f50c34b6p+0. 32 33 0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is 34 undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP: 35 acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3 36 want 0x1.e4d9bd20684f6p-3. */ 37 double 38 acosh (double x) 39 { 40 uint64_t ix = asuint64 (x); 41 42 if (unlikely (ix >= MinusZero)) 43 return __math_invalid (x); 44 45 if (unlikely (ix >= SquareLim)) 46 return log (x) + Ln2; 47 48 if (ix >= Two) 49 return log (x + sqrt (x * x - 1)); 50 51 double xm1 = x - 1; 52 return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1)); 53 } 54 55 TEST_SIG (S, D, 1, acosh, 1.0, 10.0) 56 TEST_ULP (acosh, 2.19) 57 TEST_INTERVAL (acosh, 0, 1, 10000) 58 TEST_INTERVAL (acosh, 1, 2, 100000) 59 TEST_INTERVAL (acosh, 2, 0x1p511, 100000) 60 TEST_INTERVAL (acosh, 0x1p511, inf, 100000) 61 TEST_INTERVAL (acosh, -0, -inf, 10000) 62