99 lines
3.2 KiB
C
99 lines
3.2 KiB
C
/*
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* Double-precision vector cbrt(x) function.
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*
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* Copyright (c) 2022-2023, Arm Limited.
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* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
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*/
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#include "v_math.h"
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#include "mathlib.h"
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#include "pl_sig.h"
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#include "pl_test.h"
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#if V_SUPPORTED
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#define AbsMask 0x7fffffffffffffff
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#define TwoThirds v_f64 (0x1.5555555555555p-1)
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#define TinyBound 0x001 /* top12 (smallest_normal). */
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#define BigBound 0x7ff /* top12 (infinity). */
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#define MantissaMask v_u64 (0x000fffffffffffff)
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#define HalfExp v_u64 (0x3fe0000000000000)
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#define C(i) v_f64 (__cbrt_data.poly[i])
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#define T(i) v_lookup_f64 (__cbrt_data.table, i)
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static NOINLINE v_f64_t
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specialcase (v_f64_t x, v_f64_t y, v_u64_t special)
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{
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return v_call_f64 (cbrt, x, y, special);
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}
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/* Approximation for double-precision vector cbrt(x), using low-order polynomial
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and two Newton iterations. Greatest observed error is 1.79 ULP. Errors repeat
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according to the exponent, for instance an error observed for double value
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m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
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integer.
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__v_cbrt(0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
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want 0x1.965fe72821e99p+0. */
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VPCS_ATTR v_f64_t V_NAME (cbrt) (v_f64_t x)
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{
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v_u64_t ix = v_as_u64_f64 (x);
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v_u64_t iax = ix & AbsMask;
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v_u64_t ia12 = iax >> 52;
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/* Subnormal, +/-0 and special values. */
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v_u64_t special = v_cond_u64 ((ia12 < TinyBound) | (ia12 >= BigBound));
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/* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
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version of frexp, which gets subnormal values wrong - these have to be
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special-cased as a result. */
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v_f64_t m = v_as_f64_u64 (v_bsl_u64 (MantissaMask, iax, HalfExp));
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v_s64_t e = v_as_s64_u64 (iax >> 52) - 1022;
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/* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point for
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Newton iterations. */
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v_f64_t p_01 = v_fma_f64 (C (1), m, C (0));
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v_f64_t p_23 = v_fma_f64 (C (3), m, C (2));
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v_f64_t p = v_fma_f64 (m * m, p_23, p_01);
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/* Two iterations of Newton's method for iteratively approximating cbrt. */
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v_f64_t m_by_3 = m / 3;
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v_f64_t a = v_fma_f64 (TwoThirds, p, m_by_3 / (p * p));
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a = v_fma_f64 (TwoThirds, a, m_by_3 / (a * a));
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/* Assemble the result by the following:
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cbrt(x) = cbrt(m) * 2 ^ (e / 3).
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We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
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not necessarily a multiple of 3 we lose some information.
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Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
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Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which is
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an integer in [-2, 2], and can be looked up in the table T. Hence the
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result is assembled as:
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cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */
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v_s64_t ey = e / 3;
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v_f64_t my = a * T (v_as_u64_s64 (e % 3 + 2));
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/* Vector version of ldexp. */
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v_f64_t y = v_as_f64_u64 ((v_as_u64_s64 (ey + 1023) << 52)) * my;
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/* Copy sign. */
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y = v_as_f64_u64 (v_bsl_u64 (v_u64 (AbsMask), v_as_u64_f64 (y), ix));
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if (unlikely (v_any_u64 (special)))
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return specialcase (x, y, special);
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return y;
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}
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VPCS_ALIAS
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PL_TEST_ULP (V_NAME (cbrt), 1.30)
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PL_SIG (V, D, 1, cbrt, -10.0, 10.0)
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PL_TEST_EXPECT_FENV_ALWAYS (V_NAME (cbrt))
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PL_TEST_INTERVAL (V_NAME (cbrt), 0, inf, 1000000)
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PL_TEST_INTERVAL (V_NAME (cbrt), -0, -inf, 1000000)
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#endif
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