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/*
* Double-precision cbrt(x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
PL_SIG (S, D, 1, cbrt, -10.0, 10.0)
#define AbsMask 0x7fffffffffffffff
#define TwoThirds 0x1.5555555555555p-1
#define C(i) __cbrt_data.poly[i]
#define T(i) __cbrt_data.table[i]
/* Approximation for double-precision cbrt(x), using low-order polynomial and
two Newton iterations. Greatest observed error is 1.79 ULP. Errors repeat
according to the exponent, for instance an error observed for double value
m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
integer.
cbrt(0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
want 0x1.965fe72821e99p+0. */
double
cbrt (double x)
{
uint64_t ix = asuint64 (x);
uint64_t iax = ix & AbsMask;
uint64_t sign = ix & ~AbsMask;
if (unlikely (iax == 0 || iax == 0x7f80000000000000))
return x;
/* |x| = m * 2^e, where m is in [0.5, 1.0].
We can easily decompose x into m and e using frexp. */
int e;
double m = frexp (asdouble (iax), &e);
/* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point for
Newton iterations. */
double p_01 = fma (C (1), m, C (0));
double p_23 = fma (C (3), m, C (2));
double p = fma (p_23, m * m, p_01);
/* Two iterations of Newton's method for iteratively approximating cbrt. */
double m_by_3 = m / 3;
double a = fma (TwoThirds, p, m_by_3 / (p * p));
a = fma (TwoThirds, a, m_by_3 / (a * a));
/* Assemble the result by the following:
cbrt(x) = cbrt(m) * 2 ^ (e / 3).
Let t = (2 ^ (e / 3)) / (2 ^ round(e / 3)).
Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3.
i is an integer in [-2, 2], so t can be looked up in the table T.
Hence the result is assembled as:
cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.
Which can be done easily using ldexp. */
return asdouble (asuint64 (ldexp (a * T (2 + e % 3), e / 3)) | sign);
}
PL_TEST_ULP (cbrt, 1.30)
PL_TEST_INTERVAL (cbrt, 0, inf, 1000000)
PL_TEST_INTERVAL (cbrt, -0, -inf, 1000000)
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