67 lines
1.9 KiB
C
67 lines
1.9 KiB
C
/*
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* Double-precision acosh(x) function.
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*
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* Copyright (c) 2022-2023, Arm Limited.
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* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
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*/
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#include "math_config.h"
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#include "pl_sig.h"
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#include "pl_test.h"
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#define Ln2 (0x1.62e42fefa39efp-1)
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#define MinusZero (0x8000000000000000)
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#define SquareLim (0x5fe0000000000000) /* asuint64(0x1.0p511). */
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#define Two (0x4000000000000000) /* asuint64(2.0). */
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double
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optr_aor_log_f64 (double);
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double
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log1p (double);
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/* acosh approximation using a variety of approaches on different intervals:
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acosh(x) = ln(x + sqrt(x * x - 1)).
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x >= 2^511: We cannot square x without overflow. For huge x, sqrt(x*x - 1) is
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close enough to x that we can calculate the result by ln(2x) == ln(x) +
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ln(2). The greatest observed error in this region is 0.98 ULP:
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acosh(0x1.1b9bf42923d1dp+853) got 0x1.28066a11a7c7fp+9
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want 0x1.28066a11a7c8p+9.
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x > 2: Calculate the result directly using definition of acosh(x). Greatest
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observed error in this region is 1.33 ULP:
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acosh(0x1.1e45d14bfcfa2p+1) got 0x1.71a06f50c34b5p+0
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want 0x1.71a06f50c34b6p+0.
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0 <= x <= 2: Calculate the result using log1p. For x < 1, acosh(x) is
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undefined. For 1 <= x <= 2, the largest observed error is 2.69 ULP:
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acosh(0x1.073528248093p+0) got 0x1.e4d9bd20684f3p-3
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want 0x1.e4d9bd20684f6p-3. */
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double
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acosh (double x)
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{
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uint64_t ix = asuint64 (x);
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if (unlikely (ix >= MinusZero))
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return __math_invalid (x);
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if (unlikely (ix >= SquareLim))
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return optr_aor_log_f64 (x) + Ln2;
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if (ix >= Two)
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return optr_aor_log_f64 (x + sqrt (x * x - 1));
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double xm1 = x - 1;
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return log1p (xm1 + sqrt (2 * xm1 + xm1 * xm1));
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}
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PL_SIG (S, D, 1, acosh, 1.0, 10.0)
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PL_TEST_ULP (acosh, 2.19)
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PL_TEST_INTERVAL (acosh, 0, 1, 10000)
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PL_TEST_INTERVAL (acosh, 1, 2, 100000)
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PL_TEST_INTERVAL (acosh, 2, 0x1p511, 100000)
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PL_TEST_INTERVAL (acosh, 0x1p511, inf, 100000)
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PL_TEST_INTERVAL (acosh, -0, -inf, 10000)
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