1 /*
2 * Double-precision cbrt(x) function.
3 *
4 * Copyright (c) 2022-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "math_config.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 TEST_SIG (S, D, 1, cbrt, -10.0, 10.0)
13
14 #define AbsMask 0x7fffffffffffffff
15 #define TwoThirds 0x1.5555555555555p-1
16
17 #define C(i) __cbrt_data.poly[i]
18 #define T(i) __cbrt_data.table[i]
19
20 /* Approximation for double-precision cbrt(x), using low-order polynomial and
21 two Newton iterations. Greatest observed error is 1.79 ULP. Errors repeat
22 according to the exponent, for instance an error observed for double value
23 m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
24 integer.
25 cbrt(0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
26 want 0x1.965fe72821e99p+0. */
27 double
cbrt(double x)28 cbrt (double x)
29 {
30 uint64_t ix = asuint64 (x);
31 uint64_t iax = ix & AbsMask;
32 uint64_t sign = ix & ~AbsMask;
33
34 if (unlikely (iax == 0 || iax == 0x7ff0000000000000))
35 return x;
36
37 /* |x| = m * 2^e, where m is in [0.5, 1.0].
38 We can easily decompose x into m and e using frexp. */
39 int e;
40 double m = frexp (asdouble (iax), &e);
41
42 /* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point
43 for Newton iterations. */
44 double p_01 = fma (C (1), m, C (0));
45 double p_23 = fma (C (3), m, C (2));
46 double p = fma (p_23, m * m, p_01);
47
48 /* Two iterations of Newton's method for iteratively approximating cbrt. */
49 double m_by_3 = m / 3;
50 double a = fma (TwoThirds, p, m_by_3 / (p * p));
51 a = fma (TwoThirds, a, m_by_3 / (a * a));
52
53 /* Assemble the result by the following:
54
55 cbrt(x) = cbrt(m) * 2 ^ (e / 3).
56
57 Let t = (2 ^ (e / 3)) / (2 ^ round(e / 3)).
58
59 Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3.
60 i is an integer in [-2, 2], so t can be looked up in the table T.
61 Hence the result is assembled as:
62
63 cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.
64 Which can be done easily using ldexp. */
65 return asdouble (asuint64 (ldexp (a * T (2 + e % 3), e / 3)) | sign);
66 }
67
68 TEST_ULP (cbrt, 1.30)
69 TEST_SYM_INTERVAL (cbrt, 0, inf, 1000000)
70