/*-
* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
*
* Copyright (c) 2007-2013 Bruce D. Evans
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/**
* Implementation of the natural logarithm of x for 128-bit format.
*
* First decompose x into its base 2 representation:
*
* log(x) = log(X * 2**k), where X is in [1, 2)
* = log(X) + k * log(2).
*
* Let X = X_i + e, where X_i is the center of one of the intervals
* [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
* and X is in this interval. Then
*
* log(X) = log(X_i + e)
* = log(X_i * (1 + e / X_i))
* = log(X_i) + log(1 + e / X_i).
*
* The values log(X_i) are tabulated below. Let d = e / X_i and use
*
* log(1 + d) = p(d)
*
* where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
* suitably high degree.
*
* To get sufficiently small roundoff errors, k * log(2), log(X_i), and
* sometimes (if |k| is not large) the first term in p(d) must be evaluated
* and added up in extra precision. Extra precision is not needed for the
* rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
* error is controlled mainly by the error in the second term in p(d). The
* error in this term itself is at most 0.5 ulps from the d*d operation in
* it. The error in this term relative to the first term is thus at most
* 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
* at most twice this at the point of the final rounding step. Thus the
* final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
* testing of a float variant of this function showed a maximum final error
* of 0.5008 ulps. Non-exhaustive testing of a double variant of this
* function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
*
* We made the maximum of |d| (and thus the total relative error and the
* degree of p(d)) small by using a large number of intervals. Using
* centers of intervals instead of endpoints reduces this maximum by a
* factor of 2 for a given number of intervals. p(d) is special only
* in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
* naturally. The most accurate minimax polynomial of a given degree might
* be different, but then we wouldn't want it since we would have to do
* extra work to avoid roundoff error (especially for P0*d instead of d).
*/
#ifdef DEBUG
#include <assert.h>
#include <fenv.h>
#endif
#include "fpmath.h"
#include "math.h"
#ifndef NO_STRUCT_RETURN
#define STRUCT_RETURN
#endif
#include "math_private.h"
#if !defined(NO_UTAB) && !defined(NO_UTABL)
#define USE_UTAB
#endif
/*
* Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]:
* |log(1 + d)/d - p(d)| < 2**-122.7
*/
static const long double
P2 = -0.5L,
P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */
P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */
P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */
P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */
P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */
P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */
/* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */
static const double
P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */
P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */
P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */
P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */
P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */
P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */
static volatile const double zero = 0;
#define INTERVALS 128
#define LOG2_INTERVALS 7
#define TSIZE (INTERVALS + 1)
#define G(i) (T[(i)].G)
#define F_hi(i) (T[(i)].F_hi)
#define F_lo(i) (T[(i)].F_lo)
#define ln2_hi F_hi(TSIZE - 1)
#define ln2_lo F_lo(TSIZE - 1)
#define E(i) (U[(i)].E)
#define H(i) (U[(i)].H)
static const struct {
float G; /* 1/(1 + i/128) rounded to 8/9 bits */
float F_hi; /* log(1 / G_i) rounded (see below) */
/* The compiler will insert 8 bytes of padding here. */
long double F_lo; /* next 113 bits for log(1 / G_i) */
} T[TSIZE] = {
/*
* ln2_hi and each F_hi(i) are rounded to a number of bits that
* makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
*
* The last entry (for X just below 2) is used to define ln2_hi
* and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
* with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
* This is needed for accuracy when x is just below 1. (To avoid
* special cases, such x are "reduced" strangely to X just below
* 2 and dk = -1, and then the exact cancellation is needed
* because any the error from any non-exactness would be too
* large).
*
* The relevant range of dk is [-16445, 16383]. The maximum number
* of bits in F_hi(i) that works is very dependent on i but has
* a minimum of 93. We only need about 12 bits in F_hi(i) for
* it to provide enough extra precision.
*
* We round F_hi(i) to 24 bits so that it can have type float,
* mainly to minimize the size of the table. Using all 24 bits
* in a float for it automatically satisfies the above constraints.
*/
0x800000.0p-23, 0, 0,
0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L,
0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L,
0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L,
0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L,
0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L,
0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L,
0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L,
0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L,
0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L,
0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L,
0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L,
0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L,
0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L,
0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L,
0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L,
0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L,
0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L,
0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L,
0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L,
0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L,
0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L,
0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L,
0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L,
0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L,
0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L,
0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L,
0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L,
0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L,
0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L,
0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L,
0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L,
0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L,
0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L,
0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L,
0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L,
0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L,
0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L,
0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L,
0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L,
0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L,
0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L,
0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L,
0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L,
0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L,
0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L,
0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L,
0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L,
0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L,
0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L,
0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L,
0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L,
0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L,
0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L,
0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L,
0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L,
0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L,
0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L,
0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L,
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,
0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L,
0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L,
0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L,
0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L,
0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L,
0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L,
0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L,
0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L,
0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L,
0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L,
0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L,
0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L,
0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L,
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 were very slow on
* the old sparc64 and unknown on the newer aarch64 and riscv
* machines. 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 */