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71 lines
2.1 KiB
C
71 lines
2.1 KiB
C
/*
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* Double-precision cbrt(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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PL_SIG (S, D, 1, cbrt, -10.0, 10.0)
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#define AbsMask 0x7fffffffffffffff
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#define TwoThirds 0x1.5555555555555p-1
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#define C(i) __cbrt_data.poly[i]
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#define T(i) __cbrt_data.table[i]
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/* Approximation for double-precision cbrt(x), using low-order polynomial and
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two Newton iterations. Greatest observed error is 1.79 ULP. Errors repeat
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according to the exponent, for instance an error observed for double value
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m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
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integer.
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cbrt(0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
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want 0x1.965fe72821e99p+0. */
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double
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cbrt (double x)
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{
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uint64_t ix = asuint64 (x);
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uint64_t iax = ix & AbsMask;
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uint64_t sign = ix & ~AbsMask;
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if (unlikely (iax == 0 || iax == 0x7f80000000000000))
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return x;
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/* |x| = m * 2^e, where m is in [0.5, 1.0].
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We can easily decompose x into m and e using frexp. */
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int e;
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double m = frexp (asdouble (iax), &e);
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/* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point for
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Newton iterations. */
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double p_01 = fma (C (1), m, C (0));
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double p_23 = fma (C (3), m, C (2));
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double p = fma (p_23, m * m, p_01);
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/* Two iterations of Newton's method for iteratively approximating cbrt. */
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double m_by_3 = m / 3;
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double a = fma (TwoThirds, p, m_by_3 / (p * p));
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a = fma (TwoThirds, a, m_by_3 / (a * a));
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/* Assemble the result by the following:
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cbrt(x) = cbrt(m) * 2 ^ (e / 3).
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Let t = (2 ^ (e / 3)) / (2 ^ round(e / 3)).
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Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3.
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i is an integer in [-2, 2], so t can be looked up in the table T.
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Hence the result is assembled as:
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cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.
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Which can be done easily using ldexp. */
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return asdouble (asuint64 (ldexp (a * T (2 + e % 3), e / 3)) | sign);
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}
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PL_TEST_ULP (cbrt, 1.30)
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PL_TEST_INTERVAL (cbrt, 0, inf, 1000000)
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PL_TEST_INTERVAL (cbrt, -0, -inf, 1000000)
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