260ba0bff1
Mainly focus on files that use BSD 2-Clause license, however the tool I was using mis-identified many licenses so this was mostly a manual - error prone - task. The Software Package Data Exchange (SPDX) group provides a specification to make it easier for automated tools to detect and summarize well known opensource licenses. We are gradually adopting the specification, noting that the tags are considered only advisory and do not, in any way, superceed or replace the license texts.
740 lines
30 KiB
C
740 lines
30 KiB
C
/*-
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* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
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*
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* Copyright (c) 2007-2013 Bruce D. Evans
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* All rights reserved.
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*
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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 unmodified, this list of conditions, and the following
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* 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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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/**
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* Implementation of the natural logarithm of x for 128-bit format.
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*
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* First decompose x into its base 2 representation:
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*
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* log(x) = log(X * 2**k), where X is in [1, 2)
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* = log(X) + k * log(2).
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*
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* Let X = X_i + e, where X_i is the center of one of the intervals
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* [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
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* and X is in this interval. Then
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*
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* log(X) = log(X_i + e)
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* = log(X_i * (1 + e / X_i))
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* = log(X_i) + log(1 + e / X_i).
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*
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* The values log(X_i) are tabulated below. Let d = e / X_i and use
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*
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* log(1 + d) = p(d)
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*
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* where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
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* suitably high degree.
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*
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* To get sufficiently small roundoff errors, k * log(2), log(X_i), and
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* sometimes (if |k| is not large) the first term in p(d) must be evaluated
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* and added up in extra precision. Extra precision is not needed for the
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* rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
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* error is controlled mainly by the error in the second term in p(d). The
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* error in this term itself is at most 0.5 ulps from the d*d operation in
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* it. The error in this term relative to the first term is thus at most
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* 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
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* at most twice this at the point of the final rounding step. Thus the
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* final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
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* testing of a float variant of this function showed a maximum final error
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* of 0.5008 ulps. Non-exhaustive testing of a double variant of this
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* function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
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*
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* We made the maximum of |d| (and thus the total relative error and the
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* degree of p(d)) small by using a large number of intervals. Using
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* centers of intervals instead of endpoints reduces this maximum by a
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* factor of 2 for a given number of intervals. p(d) is special only
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* in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
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* naturally. The most accurate minimax polynomial of a given degree might
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* be different, but then we wouldn't want it since we would have to do
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* extra work to avoid roundoff error (especially for P0*d instead of d).
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*/
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#ifdef DEBUG
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#include <assert.h>
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#include <fenv.h>
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#endif
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#include "fpmath.h"
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#include "math.h"
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#ifndef NO_STRUCT_RETURN
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#define STRUCT_RETURN
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#endif
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#include "math_private.h"
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#if !defined(NO_UTAB) && !defined(NO_UTABL)
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#define USE_UTAB
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#endif
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/*
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* Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]:
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* |log(1 + d)/d - p(d)| < 2**-122.7
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*/
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static const long double
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P2 = -0.5L,
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P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */
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P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */
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P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */
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P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */
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P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */
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P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */
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/* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */
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static const double
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P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */
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P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */
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P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */
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P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */
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P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */
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P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */
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static volatile const double zero = 0;
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#define INTERVALS 128
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#define LOG2_INTERVALS 7
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#define TSIZE (INTERVALS + 1)
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#define G(i) (T[(i)].G)
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#define F_hi(i) (T[(i)].F_hi)
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#define F_lo(i) (T[(i)].F_lo)
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#define ln2_hi F_hi(TSIZE - 1)
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#define ln2_lo F_lo(TSIZE - 1)
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#define E(i) (U[(i)].E)
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#define H(i) (U[(i)].H)
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static const struct {
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float G; /* 1/(1 + i/128) rounded to 8/9 bits */
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float F_hi; /* log(1 / G_i) rounded (see below) */
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/* The compiler will insert 8 bytes of padding here. */
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long double F_lo; /* next 113 bits for log(1 / G_i) */
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} T[TSIZE] = {
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/*
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* ln2_hi and each F_hi(i) are rounded to a number of bits that
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* makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
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*
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* The last entry (for X just below 2) is used to define ln2_hi
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* and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
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* with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
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* This is needed for accuracy when x is just below 1. (To avoid
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* special cases, such x are "reduced" strangely to X just below
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* 2 and dk = -1, and then the exact cancellation is needed
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* because any the error from any non-exactness would be too
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* large).
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*
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* The relevant range of dk is [-16445, 16383]. The maximum number
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* of bits in F_hi(i) that works is very dependent on i but has
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* a minimum of 93. We only need about 12 bits in F_hi(i) for
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* it to provide enough extra precision.
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*
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* We round F_hi(i) to 24 bits so that it can have type float,
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* mainly to minimize the size of the table. Using all 24 bits
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* in a float for it automatically satisfies the above constraints.
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*/
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0x800000.0p-23, 0, 0,
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0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L,
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0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L,
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0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L,
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0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L,
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0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L,
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0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L,
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0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L,
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0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L,
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0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L,
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0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L,
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0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L,
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0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L,
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0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L,
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0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L,
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0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L,
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0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L,
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0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L,
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0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L,
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0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L,
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0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L,
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0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L,
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0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L,
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0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L,
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0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L,
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0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L,
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0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L,
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0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L,
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0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L,
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0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L,
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0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L,
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0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L,
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0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L,
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0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L,
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0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L,
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0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L,
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0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L,
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0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L,
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0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L,
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0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L,
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0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L,
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0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L,
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0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L,
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0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L,
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0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L,
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0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L,
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0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L,
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0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L,
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0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L,
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0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L,
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0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L,
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0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L,
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0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L,
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0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L,
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0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L,
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0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L,
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0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L,
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0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L,
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0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L,
|
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0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L,
|
|
0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L,
|
|
0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L,
|
|
0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L,
|
|
0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L,
|
|
0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L,
|
|
0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L,
|
|
0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L,
|
|
0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L,
|
|
0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L,
|
|
0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L,
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|
0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L,
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0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L,
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0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L,
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0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L,
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0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L,
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0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L,
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0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L,
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0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L,
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0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L,
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0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L,
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0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L,
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0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L,
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0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L,
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0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L,
|
|
0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L,
|
|
0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L,
|
|
0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L,
|
|
0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L,
|
|
0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L,
|
|
0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L,
|
|
0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L,
|
|
0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L,
|
|
0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L,
|
|
0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L,
|
|
0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L,
|
|
0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L,
|
|
0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L,
|
|
0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L,
|
|
0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L,
|
|
0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L,
|
|
0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L,
|
|
0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L,
|
|
0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L,
|
|
0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L,
|
|
0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L,
|
|
0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L,
|
|
0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L,
|
|
0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L,
|
|
0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L,
|
|
0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L,
|
|
0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L,
|
|
0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L,
|
|
0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L,
|
|
0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L,
|
|
0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L,
|
|
0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L,
|
|
0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L,
|
|
0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L,
|
|
0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L,
|
|
0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L,
|
|
0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L,
|
|
0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L,
|
|
0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L,
|
|
0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L,
|
|
0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L,
|
|
0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L,
|
|
0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L,
|
|
0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L,
|
|
0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L,
|
|
};
|
|
|
|
#ifdef USE_UTAB
|
|
static const struct {
|
|
float H; /* 1 + i/INTERVALS (exact) */
|
|
float E; /* H(i) * G(i) - 1 (exact) */
|
|
} U[TSIZE] = {
|
|
0x800000.0p-23, 0,
|
|
0x810000.0p-23, -0x800000.0p-37,
|
|
0x820000.0p-23, -0x800000.0p-35,
|
|
0x830000.0p-23, -0x900000.0p-34,
|
|
0x840000.0p-23, -0x800000.0p-33,
|
|
0x850000.0p-23, -0xc80000.0p-33,
|
|
0x860000.0p-23, -0xa00000.0p-36,
|
|
0x870000.0p-23, 0x940000.0p-33,
|
|
0x880000.0p-23, 0x800000.0p-35,
|
|
0x890000.0p-23, -0xc80000.0p-34,
|
|
0x8a0000.0p-23, 0xe00000.0p-36,
|
|
0x8b0000.0p-23, 0x900000.0p-33,
|
|
0x8c0000.0p-23, -0x800000.0p-35,
|
|
0x8d0000.0p-23, -0xe00000.0p-33,
|
|
0x8e0000.0p-23, 0x880000.0p-33,
|
|
0x8f0000.0p-23, -0xa80000.0p-34,
|
|
0x900000.0p-23, -0x800000.0p-35,
|
|
0x910000.0p-23, 0x800000.0p-37,
|
|
0x920000.0p-23, 0x900000.0p-35,
|
|
0x930000.0p-23, 0xd00000.0p-35,
|
|
0x940000.0p-23, 0xe00000.0p-35,
|
|
0x950000.0p-23, 0xc00000.0p-35,
|
|
0x960000.0p-23, 0xe00000.0p-36,
|
|
0x970000.0p-23, -0x800000.0p-38,
|
|
0x980000.0p-23, -0xc00000.0p-35,
|
|
0x990000.0p-23, -0xd00000.0p-34,
|
|
0x9a0000.0p-23, 0x880000.0p-33,
|
|
0x9b0000.0p-23, 0xe80000.0p-35,
|
|
0x9c0000.0p-23, -0x800000.0p-35,
|
|
0x9d0000.0p-23, 0xb40000.0p-33,
|
|
0x9e0000.0p-23, 0x880000.0p-34,
|
|
0x9f0000.0p-23, -0xe00000.0p-35,
|
|
0xa00000.0p-23, 0x800000.0p-33,
|
|
0xa10000.0p-23, -0x900000.0p-36,
|
|
0xa20000.0p-23, -0xb00000.0p-33,
|
|
0xa30000.0p-23, -0xa00000.0p-36,
|
|
0xa40000.0p-23, 0x800000.0p-33,
|
|
0xa50000.0p-23, -0xf80000.0p-35,
|
|
0xa60000.0p-23, 0x880000.0p-34,
|
|
0xa70000.0p-23, -0x900000.0p-33,
|
|
0xa80000.0p-23, -0x800000.0p-35,
|
|
0xa90000.0p-23, 0x900000.0p-34,
|
|
0xaa0000.0p-23, 0xa80000.0p-33,
|
|
0xab0000.0p-23, -0xac0000.0p-34,
|
|
0xac0000.0p-23, -0x800000.0p-37,
|
|
0xad0000.0p-23, 0xf80000.0p-35,
|
|
0xae0000.0p-23, 0xf80000.0p-34,
|
|
0xaf0000.0p-23, -0xac0000.0p-33,
|
|
0xb00000.0p-23, -0x800000.0p-33,
|
|
0xb10000.0p-23, -0xb80000.0p-34,
|
|
0xb20000.0p-23, -0x800000.0p-34,
|
|
0xb30000.0p-23, -0xb00000.0p-35,
|
|
0xb40000.0p-23, -0x800000.0p-35,
|
|
0xb50000.0p-23, -0xe00000.0p-36,
|
|
0xb60000.0p-23, -0x800000.0p-35,
|
|
0xb70000.0p-23, -0xb00000.0p-35,
|
|
0xb80000.0p-23, -0x800000.0p-34,
|
|
0xb90000.0p-23, -0xb80000.0p-34,
|
|
0xba0000.0p-23, -0x800000.0p-33,
|
|
0xbb0000.0p-23, -0xac0000.0p-33,
|
|
0xbc0000.0p-23, 0x980000.0p-33,
|
|
0xbd0000.0p-23, 0xbc0000.0p-34,
|
|
0xbe0000.0p-23, 0xe00000.0p-36,
|
|
0xbf0000.0p-23, -0xb80000.0p-35,
|
|
0xc00000.0p-23, -0x800000.0p-33,
|
|
0xc10000.0p-23, 0xa80000.0p-33,
|
|
0xc20000.0p-23, 0x900000.0p-34,
|
|
0xc30000.0p-23, -0x800000.0p-35,
|
|
0xc40000.0p-23, -0x900000.0p-33,
|
|
0xc50000.0p-23, 0x820000.0p-33,
|
|
0xc60000.0p-23, 0x800000.0p-38,
|
|
0xc70000.0p-23, -0x820000.0p-33,
|
|
0xc80000.0p-23, 0x800000.0p-33,
|
|
0xc90000.0p-23, -0xa00000.0p-36,
|
|
0xca0000.0p-23, -0xb00000.0p-33,
|
|
0xcb0000.0p-23, 0x840000.0p-34,
|
|
0xcc0000.0p-23, -0xd00000.0p-34,
|
|
0xcd0000.0p-23, 0x800000.0p-33,
|
|
0xce0000.0p-23, -0xe00000.0p-35,
|
|
0xcf0000.0p-23, 0xa60000.0p-33,
|
|
0xd00000.0p-23, -0x800000.0p-35,
|
|
0xd10000.0p-23, 0xb40000.0p-33,
|
|
0xd20000.0p-23, -0x800000.0p-35,
|
|
0xd30000.0p-23, 0xaa0000.0p-33,
|
|
0xd40000.0p-23, -0xe00000.0p-35,
|
|
0xd50000.0p-23, 0x880000.0p-33,
|
|
0xd60000.0p-23, -0xd00000.0p-34,
|
|
0xd70000.0p-23, 0x9c0000.0p-34,
|
|
0xd80000.0p-23, -0xb00000.0p-33,
|
|
0xd90000.0p-23, -0x800000.0p-38,
|
|
0xda0000.0p-23, 0xa40000.0p-33,
|
|
0xdb0000.0p-23, -0xdc0000.0p-34,
|
|
0xdc0000.0p-23, 0xc00000.0p-35,
|
|
0xdd0000.0p-23, 0xca0000.0p-33,
|
|
0xde0000.0p-23, -0xb80000.0p-34,
|
|
0xdf0000.0p-23, 0xd00000.0p-35,
|
|
0xe00000.0p-23, 0xc00000.0p-33,
|
|
0xe10000.0p-23, -0xf40000.0p-34,
|
|
0xe20000.0p-23, 0x800000.0p-37,
|
|
0xe30000.0p-23, 0x860000.0p-33,
|
|
0xe40000.0p-23, -0xc80000.0p-33,
|
|
0xe50000.0p-23, -0xa80000.0p-34,
|
|
0xe60000.0p-23, 0xe00000.0p-36,
|
|
0xe70000.0p-23, 0x880000.0p-33,
|
|
0xe80000.0p-23, -0xe00000.0p-33,
|
|
0xe90000.0p-23, -0xfc0000.0p-34,
|
|
0xea0000.0p-23, -0x800000.0p-35,
|
|
0xeb0000.0p-23, 0xe80000.0p-35,
|
|
0xec0000.0p-23, 0x900000.0p-33,
|
|
0xed0000.0p-23, 0xe20000.0p-33,
|
|
0xee0000.0p-23, -0xac0000.0p-33,
|
|
0xef0000.0p-23, -0xc80000.0p-34,
|
|
0xf00000.0p-23, -0x800000.0p-35,
|
|
0xf10000.0p-23, 0x800000.0p-35,
|
|
0xf20000.0p-23, 0xb80000.0p-34,
|
|
0xf30000.0p-23, 0x940000.0p-33,
|
|
0xf40000.0p-23, 0xc80000.0p-33,
|
|
0xf50000.0p-23, -0xf20000.0p-33,
|
|
0xf60000.0p-23, -0xc80000.0p-33,
|
|
0xf70000.0p-23, -0xa20000.0p-33,
|
|
0xf80000.0p-23, -0x800000.0p-33,
|
|
0xf90000.0p-23, -0xc40000.0p-34,
|
|
0xfa0000.0p-23, -0x900000.0p-34,
|
|
0xfb0000.0p-23, -0xc80000.0p-35,
|
|
0xfc0000.0p-23, -0x800000.0p-35,
|
|
0xfd0000.0p-23, -0x900000.0p-36,
|
|
0xfe0000.0p-23, -0x800000.0p-37,
|
|
0xff0000.0p-23, -0x800000.0p-39,
|
|
0x800000.0p-22, 0,
|
|
};
|
|
#endif /* USE_UTAB */
|
|
|
|
#ifdef STRUCT_RETURN
|
|
#define RETURN1(rp, v) do { \
|
|
(rp)->hi = (v); \
|
|
(rp)->lo_set = 0; \
|
|
return; \
|
|
} while (0)
|
|
|
|
#define RETURN2(rp, h, l) do { \
|
|
(rp)->hi = (h); \
|
|
(rp)->lo = (l); \
|
|
(rp)->lo_set = 1; \
|
|
return; \
|
|
} while (0)
|
|
|
|
struct ld {
|
|
long double hi;
|
|
long double lo;
|
|
int lo_set;
|
|
};
|
|
#else
|
|
#define RETURN1(rp, v) RETURNF(v)
|
|
#define RETURN2(rp, h, l) RETURNI((h) + (l))
|
|
#endif
|
|
|
|
#ifdef STRUCT_RETURN
|
|
static inline __always_inline void
|
|
k_logl(long double x, struct ld *rp)
|
|
#else
|
|
long double
|
|
logl(long double x)
|
|
#endif
|
|
{
|
|
long double d, val_hi, val_lo;
|
|
double dd, dk;
|
|
uint64_t lx, llx;
|
|
int i, k;
|
|
uint16_t hx;
|
|
|
|
EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
|
|
k = -16383;
|
|
#if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
|
|
if (x == 1)
|
|
RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
|
|
#endif
|
|
if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
|
|
if (((hx & 0x7fff) | lx | llx) == 0)
|
|
RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
|
|
if (hx != 0)
|
|
/* log(neg or NaN) = qNaN: */
|
|
RETURN1(rp, (x - x) / zero);
|
|
x *= 0x1.0p113; /* subnormal; scale up x */
|
|
EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
|
|
k = -16383 - 113;
|
|
} else if (hx >= 0x7fff)
|
|
RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
|
|
#ifndef STRUCT_RETURN
|
|
ENTERI();
|
|
#endif
|
|
k += hx;
|
|
dk = k;
|
|
|
|
/* Scale x to be in [1, 2). */
|
|
SET_LDBL_EXPSIGN(x, 0x3fff);
|
|
|
|
/* 0 <= i <= INTERVALS: */
|
|
#define L2I (49 - LOG2_INTERVALS)
|
|
i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
|
|
|
|
/*
|
|
* -0.005280 < d < 0.004838. In particular, the infinite-
|
|
* precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
|
|
* ensures that d is representable without extra precision for
|
|
* this bound on |d| (since when this calculation is expressed
|
|
* as x*G(i)-1, the multiplication needs as many extra bits as
|
|
* G(i) has and the subtraction cancels 8 bits). But for
|
|
* most i (107 cases out of 129), the infinite-precision |d|
|
|
* is <= 2**-8. G(i) is rounded to 9 bits for such i to give
|
|
* better accuracy (this works by improving the bound on |d|,
|
|
* which in turn allows rounding to 9 bits in more cases).
|
|
* This is only important when the original x is near 1 -- it
|
|
* lets us avoid using a special method to give the desired
|
|
* accuracy for such x.
|
|
*/
|
|
if (0)
|
|
d = x * G(i) - 1;
|
|
else {
|
|
#ifdef USE_UTAB
|
|
d = (x - H(i)) * G(i) + E(i);
|
|
#else
|
|
long double x_hi;
|
|
double x_lo;
|
|
|
|
/*
|
|
* Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
|
|
* G(i) has at most 9 bits, so the splitting point is not
|
|
* critical.
|
|
*/
|
|
INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
|
|
llx & 0xffffffffff000000ULL);
|
|
x_lo = x - x_hi;
|
|
d = x_hi * G(i) - 1 + x_lo * G(i);
|
|
#endif
|
|
}
|
|
|
|
/*
|
|
* Our algorithm depends on exact cancellation of F_lo(i) and
|
|
* F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
|
|
* at the end of the table. This and other technical complications
|
|
* make it difficult to avoid the double scaling in (dk*ln2) *
|
|
* log(base) for base != e without losing more accuracy and/or
|
|
* efficiency than is gained.
|
|
*/
|
|
/*
|
|
* Use double precision operations wherever possible, since long
|
|
* double operations are emulated and are very slow on the only
|
|
* known machines that support ld128 (sparc64). Also, don't try
|
|
* to improve parallelism by increasing the number of operations,
|
|
* since any parallelism on such machines is needed for the
|
|
* emulation. Horner's method is good for this, and is also good
|
|
* for accuracy. Horner's method doesn't handle the `lo' term
|
|
* well, either for efficiency or accuracy. However, for accuracy
|
|
* we evaluate d * d * P2 separately to take advantage of
|
|
* by P2 being exact, and this gives a good place to sum the 'lo'
|
|
* term too.
|
|
*/
|
|
dd = (double)d;
|
|
val_lo = d * d * d * (P3 +
|
|
d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
|
|
dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
|
|
dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2;
|
|
val_hi = d;
|
|
#ifdef DEBUG
|
|
if (fetestexcept(FE_UNDERFLOW))
|
|
breakpoint();
|
|
#endif
|
|
|
|
_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
|
|
RETURN2(rp, val_hi, val_lo);
|
|
}
|
|
|
|
long double
|
|
log1pl(long double x)
|
|
{
|
|
long double d, d_hi, f_lo, val_hi, val_lo;
|
|
long double f_hi, twopminusk;
|
|
double d_lo, dd, dk;
|
|
uint64_t lx, llx;
|
|
int i, k;
|
|
int16_t ax, hx;
|
|
|
|
DOPRINT_START(&x);
|
|
EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
|
|
if (hx < 0x3fff) { /* x < 1, or x neg NaN */
|
|
ax = hx & 0x7fff;
|
|
if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
|
|
if (ax == 0x3fff && (lx | llx) == 0)
|
|
RETURNP(-1 / zero); /* log1p(-1) = -Inf */
|
|
/* log1p(x < 1, or x NaN) = qNaN: */
|
|
RETURNP((x - x) / (x - x));
|
|
}
|
|
if (ax <= 0x3f8d) { /* |x| < 2**-113 */
|
|
if ((int)x == 0)
|
|
RETURNP(x); /* x with inexact if x != 0 */
|
|
}
|
|
f_hi = 1;
|
|
f_lo = x;
|
|
} else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
|
|
RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
|
|
} else if (hx < 0x40e1) { /* 1 <= x < 2**226 */
|
|
f_hi = x;
|
|
f_lo = 1;
|
|
} else { /* 2**226 <= x < +Inf */
|
|
f_hi = x;
|
|
f_lo = 0; /* avoid underflow of the P3 term */
|
|
}
|
|
ENTERI();
|
|
x = f_hi + f_lo;
|
|
f_lo = (f_hi - x) + f_lo;
|
|
|
|
EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
|
|
k = -16383;
|
|
|
|
k += hx;
|
|
dk = k;
|
|
|
|
SET_LDBL_EXPSIGN(x, 0x3fff);
|
|
twopminusk = 1;
|
|
SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
|
|
f_lo *= twopminusk;
|
|
|
|
i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
|
|
|
|
/*
|
|
* x*G(i)-1 (with a reduced x) can be represented exactly, as
|
|
* above, but now we need to evaluate the polynomial on d =
|
|
* (x+f_lo)*G(i)-1 and extra precision is needed for that.
|
|
* Since x+x_lo is a hi+lo decomposition and subtracting 1
|
|
* doesn't lose too many bits, an inexact calculation for
|
|
* f_lo*G(i) is good enough.
|
|
*/
|
|
if (0)
|
|
d_hi = x * G(i) - 1;
|
|
else {
|
|
#ifdef USE_UTAB
|
|
d_hi = (x - H(i)) * G(i) + E(i);
|
|
#else
|
|
long double x_hi;
|
|
double x_lo;
|
|
|
|
INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
|
|
llx & 0xffffffffff000000ULL);
|
|
x_lo = x - x_hi;
|
|
d_hi = x_hi * G(i) - 1 + x_lo * G(i);
|
|
#endif
|
|
}
|
|
d_lo = f_lo * G(i);
|
|
|
|
/*
|
|
* This is _2sumF(d_hi, d_lo) inlined. The condition
|
|
* (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
|
|
* always satisifed, so it is not clear that this works, but
|
|
* it works in practice. It works even if it gives a wrong
|
|
* normalized d_lo, since |d_lo| > |d_hi| implies that i is
|
|
* nonzero and d is tiny, so the F(i) term dominates d_lo.
|
|
* In float precision:
|
|
* (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
|
|
* And if d is only a little tinier than that, we would have
|
|
* another underflow problem for the P3 term; this is also ruled
|
|
* out by exhaustive testing.)
|
|
*/
|
|
d = d_hi + d_lo;
|
|
d_lo = d_hi - d + d_lo;
|
|
d_hi = d;
|
|
|
|
dd = (double)d;
|
|
val_lo = d * d * d * (P3 +
|
|
d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
|
|
dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
|
|
dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2;
|
|
val_hi = d_hi;
|
|
#ifdef DEBUG
|
|
if (fetestexcept(FE_UNDERFLOW))
|
|
breakpoint();
|
|
#endif
|
|
|
|
_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
|
|
RETURN2PI(val_hi, val_lo);
|
|
}
|
|
|
|
#ifdef STRUCT_RETURN
|
|
|
|
long double
|
|
logl(long double x)
|
|
{
|
|
struct ld r;
|
|
|
|
ENTERI();
|
|
DOPRINT_START(&x);
|
|
k_logl(x, &r);
|
|
RETURNSPI(&r);
|
|
}
|
|
|
|
/*
|
|
* 29+113 bit decompositions. The bits are distributed so that the products
|
|
* of the hi terms are exact in double precision. The types are chosen so
|
|
* that the products of the hi terms are done in at least double precision,
|
|
* without any explicit conversions. More natural choices would require a
|
|
* slow long double precision multiplication.
|
|
*/
|
|
static const double
|
|
invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */
|
|
invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */
|
|
static const long double
|
|
invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */
|
|
invln2_lo = 6.33178418956604368501892137426645911e-10L; /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
|
|
|
|
long double
|
|
log10l(long double x)
|
|
{
|
|
struct ld r;
|
|
long double lo;
|
|
float hi;
|
|
|
|
ENTERI();
|
|
DOPRINT_START(&x);
|
|
k_logl(x, &r);
|
|
if (!r.lo_set)
|
|
RETURNPI(r.hi);
|
|
_2sumF(r.hi, r.lo);
|
|
hi = r.hi;
|
|
lo = r.lo + (r.hi - hi);
|
|
RETURN2PI(invln10_hi * hi,
|
|
(invln10_lo + invln10_hi) * lo + invln10_lo * hi);
|
|
}
|
|
|
|
long double
|
|
log2l(long double x)
|
|
{
|
|
struct ld r;
|
|
long double lo;
|
|
float hi;
|
|
|
|
ENTERI();
|
|
DOPRINT_START(&x);
|
|
k_logl(x, &r);
|
|
if (!r.lo_set)
|
|
RETURNPI(r.hi);
|
|
_2sumF(r.hi, r.lo);
|
|
hi = r.hi;
|
|
lo = r.lo + (r.hi - hi);
|
|
RETURN2PI(invln2_hi * hi,
|
|
(invln2_lo + invln2_hi) * lo + invln2_lo * hi);
|
|
}
|
|
|
|
#endif /* STRUCT_RETURN */
|