159 lines
5.1 KiB
C
159 lines
5.1 KiB
C
/* Double-precision log2(x) function.
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Copyright (c) 2018 Arm Ltd. All rights reserved.
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SPDX-License-Identifier: BSD-3-Clause
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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1. Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. The name of the company may not be used to endorse or promote
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products derived from this software without specific prior written
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permission.
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THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS'' IS AND ANY EXPRESS OR IMPLIED
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WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
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MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
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TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
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#include "fdlibm.h"
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#if !__OBSOLETE_MATH
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#include <math.h>
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#include <stdint.h>
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#include "math_config.h"
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#define T __log2_data.tab
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#define T2 __log2_data.tab2
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#define B __log2_data.poly1
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#define A __log2_data.poly
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#define InvLn2hi __log2_data.invln2hi
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#define InvLn2lo __log2_data.invln2lo
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#define N (1 << LOG2_TABLE_BITS)
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#define OFF 0x3fe6000000000000
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/* Top 16 bits of a double. */
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static inline uint32_t
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top16 (double x)
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{
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return asuint64 (x) >> 48;
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}
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double
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(log2) (double x)
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{
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
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uint64_t ix, iz, tmp;
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uint32_t top;
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int k, i;
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ix = asuint64 (x);
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top = top16 (x);
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#if LOG2_POLY1_ORDER == 11
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# define LO asuint64 (1.0 - 0x1.5b51p-5)
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# define HI asuint64 (1.0 + 0x1.6ab2p-5)
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#endif
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if (unlikely (ix - LO < HI - LO))
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{
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/* Handle close to 1.0 inputs separately. */
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/* Fix sign of zero with downward rounding when x==1. */
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if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
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return 0;
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r = x - 1.0;
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#if HAVE_FAST_FMA
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hi = r * InvLn2hi;
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lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);
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#else
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double_t rhi, rlo;
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rhi = asdouble (asuint64 (r) & -1ULL << 32);
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rlo = r - rhi;
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hi = rhi * InvLn2hi;
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lo = rlo * InvLn2hi + r * InvLn2lo;
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#endif
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r2 = r * r; /* rounding error: 0x1p-62. */
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r4 = r2 * r2;
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#if LOG2_POLY1_ORDER == 11
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/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
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p = r2 * (B[0] + r * B[1]);
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y = hi + p;
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lo += hi - y + p;
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lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
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+ r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
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y += lo;
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#endif
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return y;
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}
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if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
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{
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/* x < 0x1p-1022 or inf or nan. */
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if (ix * 2 == 0)
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return __math_divzero (1);
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if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
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return x;
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if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
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return __math_invalid (x);
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/* x is subnormal, normalize it. */
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ix = asuint64 (x * 0x1p52);
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ix -= 52ULL << 52;
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}
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/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
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The range is split into N subintervals.
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The ith subinterval contains z and c is near its center. */
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tmp = ix - OFF;
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i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
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k = (int64_t) tmp >> 52; /* arithmetic shift */
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iz = ix - (tmp & 0xfffULL << 52);
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invc = T[i].invc;
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logc = T[i].logc;
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z = asdouble (iz);
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kd = (double_t) k;
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/* log2(x) = log2(z/c) + log2(c) + k. */
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/* r ~= z/c - 1, |r| < 1/(2*N). */
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#if HAVE_FAST_FMA
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/* rounding error: 0x1p-55/N. */
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r = fma (z, invc, -1.0);
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t1 = r * InvLn2hi;
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t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);
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#else
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double_t rhi, rlo;
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/* rounding error: 0x1p-55/N + 0x1p-65. */
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r = (z - T2[i].chi - T2[i].clo) * invc;
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rhi = asdouble (asuint64 (r) & -1ULL << 32);
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rlo = r - rhi;
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t1 = rhi * InvLn2hi;
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t2 = rlo * InvLn2hi + r * InvLn2lo;
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#endif
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/* hi + lo = r/ln2 + log2(c) + k. */
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t3 = kd + logc;
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hi = t3 + t1;
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lo = t3 - hi + t1 + t2;
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/* log2(r+1) = r/ln2 + r^2*poly(r). */
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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r2 = r * r; /* rounding error: 0x1p-54/N^2. */
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r4 = r2 * r2;
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#if LOG2_POLY_ORDER == 7
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/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
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~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
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p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
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y = lo + r2 * p + hi;
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#endif
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return y;
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}
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#endif
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