1/*
2 * Single-precision cbrt(x) function.
3 *
4 * Copyright (c) 2022-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8#include "poly_scalar_f32.h"
9#include "math_config.h"
10#include "pl_sig.h"
11#include "pl_test.h"
12
13#define AbsMask 0x7fffffff
14#define SignMask 0x80000000
15#define TwoThirds 0x1.555556p-1f
16
17#define T(i) __cbrtf_data.table[i]
18
19/* Approximation for single-precision cbrt(x), using low-order polynomial and
20   one Newton iteration on a reduced interval. Greatest error is 1.5 ULP. This
21   is observed for every value where the mantissa is 0x1.81410e and the exponent
22   is a multiple of 3, for example:
23   cbrtf(0x1.81410ep+30) got 0x1.255d96p+10
24			want 0x1.255d92p+10.  */
25float
26cbrtf (float x)
27{
28  uint32_t ix = asuint (x);
29  uint32_t iax = ix & AbsMask;
30  uint32_t sign = ix & SignMask;
31
32  if (unlikely (iax == 0 || iax == 0x7f800000))
33    return x;
34
35  /* |x| = m * 2^e, where m is in [0.5, 1.0].
36     We can easily decompose x into m and e using frexpf.  */
37  int e;
38  float m = frexpf (asfloat (iax), &e);
39
40  /* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
41     the less accurate the next stage of the algorithm needs to be. An order-4
42     polynomial is enough for one Newton iteration.  */
43  float p = pairwise_poly_3_f32 (m, m * m, __cbrtf_data.poly);
44
45  /* One iteration of Newton's method for iteratively approximating cbrt.  */
46  float m_by_3 = m / 3;
47  float a = fmaf (TwoThirds, p, m_by_3 / (p * p));
48
49  /* Assemble the result by the following:
50
51     cbrt(x) = cbrt(m) * 2 ^ (e / 3).
52
53     Let t = (2 ^ (e / 3)) / (2 ^ round(e / 3)).
54
55     Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3.
56     i is an integer in [-2, 2], so t can be looked up in the table T.
57     Hence the result is assembled as:
58
59     cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.
60     Which can be done easily using ldexpf.  */
61  return asfloat (asuint (ldexpf (a * T (2 + e % 3), e / 3)) | sign);
62}
63
64PL_SIG (S, F, 1, cbrt, -10.0, 10.0)
65PL_TEST_ULP (cbrtf, 1.03)
66PL_TEST_SYM_INTERVAL (cbrtf, 0, inf, 1000000)
67