1 /*
2 * Single-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 "poly_scalar_f32.h"
9 #include "math_config.h"
10 #include "test_sig.h"
11 #include "test_defs.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
22 exponent is a multiple of 3, for example:
23 cbrtf(0x1.81410ep+30) got 0x1.255d96p+10
24 want 0x1.255d92p+10. */
25 float
cbrtf(float x)26 cbrtf (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
64 TEST_SIG (S, F, 1, cbrt, -10.0, 10.0)
65 TEST_ULP (cbrtf, 1.03)
66 TEST_SYM_INTERVAL (cbrtf, 0, inf, 1000000)
67