1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision cbrt(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner
8*f3087befSAndrew Turner #include "poly_scalar_f32.h"
9*f3087befSAndrew Turner #include "math_config.h"
10*f3087befSAndrew Turner #include "test_sig.h"
11*f3087befSAndrew Turner #include "test_defs.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner #define AbsMask 0x7fffffff
14*f3087befSAndrew Turner #define SignMask 0x80000000
15*f3087befSAndrew Turner #define TwoThirds 0x1.555556p-1f
16*f3087befSAndrew Turner
17*f3087befSAndrew Turner #define T(i) __cbrtf_data.table[i]
18*f3087befSAndrew Turner
19*f3087befSAndrew Turner /* Approximation for single-precision cbrt(x), using low-order polynomial and
20*f3087befSAndrew Turner one Newton iteration on a reduced interval. Greatest error is 1.5 ULP. This
21*f3087befSAndrew Turner is observed for every value where the mantissa is 0x1.81410e and the
22*f3087befSAndrew Turner exponent is a multiple of 3, for example:
23*f3087befSAndrew Turner cbrtf(0x1.81410ep+30) got 0x1.255d96p+10
24*f3087befSAndrew Turner want 0x1.255d92p+10. */
25*f3087befSAndrew Turner float
cbrtf(float x)26*f3087befSAndrew Turner cbrtf (float x)
27*f3087befSAndrew Turner {
28*f3087befSAndrew Turner uint32_t ix = asuint (x);
29*f3087befSAndrew Turner uint32_t iax = ix & AbsMask;
30*f3087befSAndrew Turner uint32_t sign = ix & SignMask;
31*f3087befSAndrew Turner
32*f3087befSAndrew Turner if (unlikely (iax == 0 || iax == 0x7f800000))
33*f3087befSAndrew Turner return x;
34*f3087befSAndrew Turner
35*f3087befSAndrew Turner /* |x| = m * 2^e, where m is in [0.5, 1.0].
36*f3087befSAndrew Turner We can easily decompose x into m and e using frexpf. */
37*f3087befSAndrew Turner int e;
38*f3087befSAndrew Turner float m = frexpf (asfloat (iax), &e);
39*f3087befSAndrew Turner
40*f3087befSAndrew Turner /* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
41*f3087befSAndrew Turner the less accurate the next stage of the algorithm needs to be. An order-4
42*f3087befSAndrew Turner polynomial is enough for one Newton iteration. */
43*f3087befSAndrew Turner float p = pairwise_poly_3_f32 (m, m * m, __cbrtf_data.poly);
44*f3087befSAndrew Turner
45*f3087befSAndrew Turner /* One iteration of Newton's method for iteratively approximating cbrt. */
46*f3087befSAndrew Turner float m_by_3 = m / 3;
47*f3087befSAndrew Turner float a = fmaf (TwoThirds, p, m_by_3 / (p * p));
48*f3087befSAndrew Turner
49*f3087befSAndrew Turner /* Assemble the result by the following:
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner cbrt(x) = cbrt(m) * 2 ^ (e / 3).
52*f3087befSAndrew Turner
53*f3087befSAndrew Turner Let t = (2 ^ (e / 3)) / (2 ^ round(e / 3)).
54*f3087befSAndrew Turner
55*f3087befSAndrew Turner Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3.
56*f3087befSAndrew Turner i is an integer in [-2, 2], so t can be looked up in the table T.
57*f3087befSAndrew Turner Hence the result is assembled as:
58*f3087befSAndrew Turner
59*f3087befSAndrew Turner cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.
60*f3087befSAndrew Turner Which can be done easily using ldexpf. */
61*f3087befSAndrew Turner return asfloat (asuint (ldexpf (a * T (2 + e % 3), e / 3)) | sign);
62*f3087befSAndrew Turner }
63*f3087befSAndrew Turner
64*f3087befSAndrew Turner TEST_SIG (S, F, 1, cbrt, -10.0, 10.0)
65*f3087befSAndrew Turner TEST_ULP (cbrtf, 1.03)
66*f3087befSAndrew Turner TEST_SYM_INTERVAL (cbrtf, 0, inf, 1000000)
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