1 /* 2 * Double-precision acosh(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 Ln2 (0x1.62e42fefa39efp-1) 13 #define MinusZero (0x8000000000000000) 14 #define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511). */ 15 #define Two (0x4000000000000000) /* asuint64(2.0). */ 16 17 double 18 optr_aor_log_f64 (double); 19 20 double 21 log1p (double); 22 23 /* acosh approximation using a variety of approaches on different intervals: 24 25 acosh(x) = ln(x + sqrt(x * x - 1)). 26 27 x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) is 28 close enough to x that we can calculate the result by ln(2x) == ln(x) + 29 ln(2). The greatest observed error in this region is 0.98 ULP: 30 acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9 31 want 0x1.28066a11a7c8p+9. 32 33 x > 2: Calculate the result directly using definition of acosh(x). Greatest 34 observed error in this region is 1.33 ULP: 35 acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0 36 want 0x1.71a06f50c34b6p+0. 37 38 0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is 39 undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP: 40 acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3 41 want 0x1.e4d9bd20684f6p-3. */ 42 double 43 acosh (double x) 44 { 45 uint64_t ix = asuint64 (x); 46 47 if (unlikely (ix >= MinusZero)) 48 return __math_invalid (x); 49 50 if (unlikely (ix >= SquareLim)) 51 return optr_aor_log_f64 (x) + Ln2; 52 53 if (ix >= Two) 54 return optr_aor_log_f64 (x + sqrt (x * x - 1)); 55 56 double xm1 = x - 1; 57 return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1)); 58 } 59 60 PL_SIG (S, D, 1, acosh, 1.0, 10.0) 61 PL_TEST_ULP (acosh, 2.19) 62 PL_TEST_INTERVAL (acosh, 0, 1, 10000) 63 PL_TEST_INTERVAL (acosh, 1, 2, 100000) 64 PL_TEST_INTERVAL (acosh, 2, 0x1p511, 100000) 65 PL_TEST_INTERVAL (acosh, 0x1p511, inf, 100000) 66 PL_TEST_INTERVAL (acosh, -0, -inf, 10000) 67