1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision acosh(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "mathlib.h" 9*f3087befSAndrew Turner #include "math_config.h" 10*f3087befSAndrew Turner #include "test_sig.h" 11*f3087befSAndrew Turner #include "test_defs.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner #define Ln2 (0x1.62e42fefa39efp-1) 14*f3087befSAndrew Turner #define MinusZero (0x8000000000000000) 15*f3087befSAndrew Turner #define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511). */ 16*f3087befSAndrew Turner #define Two (0x4000000000000000) /* asuint64(2.0). */ 17*f3087befSAndrew Turner 18*f3087befSAndrew Turner /* acosh approximation using a variety of approaches on different intervals: 19*f3087befSAndrew Turner 20*f3087befSAndrew Turner acosh(x) = ln(x + sqrt(x * x - 1)). 21*f3087befSAndrew Turner 22*f3087befSAndrew Turner x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) 23*f3087befSAndrew Turner is close enough to x that we can calculate the result by ln(2x) == ln(x) + 24*f3087befSAndrew Turner ln(2). The greatest observed error in this region is 0.98 ULP: 25*f3087befSAndrew Turner acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9 26*f3087befSAndrew Turner want 0x1.28066a11a7c8p+9. 27*f3087befSAndrew Turner 28*f3087befSAndrew Turner x > 2: Calculate the result directly using definition of acosh(x). Greatest 29*f3087befSAndrew Turner observed error in this region is 1.33 ULP: 30*f3087befSAndrew Turner acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0 31*f3087befSAndrew Turner want 0x1.71a06f50c34b6p+0. 32*f3087befSAndrew Turner 33*f3087befSAndrew Turner 0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is 34*f3087befSAndrew Turner undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP: 35*f3087befSAndrew Turner acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3 36*f3087befSAndrew Turner want 0x1.e4d9bd20684f6p-3. */ 37*f3087befSAndrew Turner double 38*f3087befSAndrew Turner acosh (double x) 39*f3087befSAndrew Turner { 40*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 41*f3087befSAndrew Turner 42*f3087befSAndrew Turner if (unlikely (ix >= MinusZero)) 43*f3087befSAndrew Turner return __math_invalid (x); 44*f3087befSAndrew Turner 45*f3087befSAndrew Turner if (unlikely (ix >= SquareLim)) 46*f3087befSAndrew Turner return log (x) + Ln2; 47*f3087befSAndrew Turner 48*f3087befSAndrew Turner if (ix >= Two) 49*f3087befSAndrew Turner return log (x + sqrt (x * x - 1)); 50*f3087befSAndrew Turner 51*f3087befSAndrew Turner double xm1 = x - 1; 52*f3087befSAndrew Turner return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1)); 53*f3087befSAndrew Turner } 54*f3087befSAndrew Turner 55*f3087befSAndrew Turner TEST_SIG (S, D, 1, acosh, 1.0, 10.0) 56*f3087befSAndrew Turner TEST_ULP (acosh, 2.19) 57*f3087befSAndrew Turner TEST_INTERVAL (acosh, 0, 1, 10000) 58*f3087befSAndrew Turner TEST_INTERVAL (acosh, 1, 2, 100000) 59*f3087befSAndrew Turner TEST_INTERVAL (acosh, 2, 0x1p511, 100000) 60*f3087befSAndrew Turner TEST_INTERVAL (acosh, 0x1p511, inf, 100000) 61*f3087befSAndrew Turner TEST_INTERVAL (acosh, -0, -inf, 10000) 62