97 lines
3.1 KiB
C
97 lines
3.1 KiB
C
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/*
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* Single-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 0x7fffffff
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#define SignMask v_u32 (0x80000000)
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#define TwoThirds v_f32 (0x1.555556p-1f)
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#define SmallestNormal 0x00800000
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#define MantissaMask 0x007fffff
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#define HalfExp 0x3f000000
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#define C(i) v_f32 (__cbrtf_data.poly[i])
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#define T(i) v_lookup_f32 (__cbrtf_data.table, i)
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static NOINLINE v_f32_t
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specialcase (v_f32_t x, v_f32_t y, v_u32_t special)
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{
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return v_call_f32 (cbrtf, x, y, special);
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}
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/* Approximation for vector single-precision cbrt(x) using Newton iteration with
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initial guess obtained by a low-order polynomial. Greatest error is 1.5 ULP.
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This is observed for every value where the mantissa is 0x1.81410e and the
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exponent is a multiple of 3, for example:
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__v_cbrtf(0x1.81410ep+30) got 0x1.255d96p+10
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want 0x1.255d92p+10. */
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VPCS_ATTR v_f32_t V_NAME (cbrtf) (v_f32_t x)
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{
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v_u32_t ix = v_as_u32_f32 (x);
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v_u32_t iax = ix & AbsMask;
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/* Subnormal, +/-0 and special values. */
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v_u32_t special = v_cond_u32 ((iax < SmallestNormal) | (iax >= 0x7f800000));
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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 frexpf, which gets subnormal values wrong - these have to be
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special-cased as a result. */
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v_f32_t m = v_as_f32_u32 ((iax & MantissaMask) | HalfExp);
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v_s32_t e = v_as_s32_u32 (iax >> 23) - 126;
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/* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
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the less accurate the next stage of the algorithm needs to be. An order-4
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polynomial is enough for one Newton iteration. */
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v_f32_t p_01 = v_fma_f32 (C (1), m, C (0));
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v_f32_t p_23 = v_fma_f32 (C (3), m, C (2));
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v_f32_t p = v_fma_f32 (m * m, p_23, p_01);
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/* One iteration of Newton's method for iteratively approximating cbrt. */
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v_f32_t m_by_3 = m / 3;
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v_f32_t a = v_fma_f32 (TwoThirds, p, m_by_3 / (p * p));
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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_s32_t ey = e / 3;
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v_f32_t my = a * T (v_as_u32_s32 (e % 3 + 2));
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/* Vector version of ldexpf. */
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v_f32_t y = v_as_f32_u32 ((v_as_u32_s32 (ey + 127) << 23)) * my;
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/* Copy sign. */
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y = v_as_f32_u32 (v_bsl_u32 (SignMask, ix, v_as_u32_f32 (y)));
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if (unlikely (v_any_u32 (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_SIG (V, F, 1, cbrt, -10.0, 10.0)
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PL_TEST_ULP (V_NAME (cbrtf), 1.03)
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PL_TEST_EXPECT_FENV_ALWAYS (V_NAME (cbrtf))
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PL_TEST_INTERVAL (V_NAME (cbrtf), 0, inf, 1000000)
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PL_TEST_INTERVAL (V_NAME (cbrtf), -0, -inf, 1000000)
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#endif
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