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Add logl, log2l, log10l, and log1pl.

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Submitted by:	bde
This commit is contained in:
David Schultz 2013-06-03 09:14:31 +00:00
parent 1e65d73c74
commit 25a4d6bfda
11 changed files with 1809 additions and 18 deletions
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8
lib/msun/Makefile
View File

@ -99,8 +99,8 @@ COMMON_SRCS+= e_acosl.c e_asinl.c e_atan2l.c e_fmodl.c \
invtrig.c k_cosl.c k_sinl.c k_tanl.c \
s_atanl.c s_cbrtl.c s_ceill.c s_cosl.c s_cprojl.c \
s_csqrtl.c s_exp2l.c s_expl.c s_floorl.c s_fmal.c \
s_frexpl.c s_logbl.c s_nanl.c s_nextafterl.c s_nexttoward.c \
s_remquol.c s_rintl.c s_scalbnl.c \
s_frexpl.c s_logbl.c s_logl.c s_nanl.c s_nextafterl.c \
s_nexttoward.c s_remquol.c s_rintl.c s_scalbnl.c \
s_sinl.c s_tanl.c s_truncl.c w_cabsl.c
.endif
@ -187,7 +187,9 @@ MLINKS+=j0.3 j1.3 j0.3 jn.3 j0.3 y0.3 j0.3 y1.3 j0.3 y1f.3 j0.3 yn.3
MLINKS+=j0.3 j0f.3 j0.3 j1f.3 j0.3 jnf.3 j0.3 y0f.3 j0.3 ynf.3
MLINKS+=lgamma.3 gamma.3 lgamma.3 gammaf.3 lgamma.3 lgammaf.3 \
lgamma.3 tgamma.3 lgamma.3 tgammaf.3
MLINKS+=log.3 log10.3 log.3 log10f.3 log.3 log1p.3 log.3 log1pf.3 log.3 logf.3 log.3 log2.3 log.3 log2f.3
MLINKS+=log.3 log10.3 log.3 log10f.3 log.3 log10l.3 log.3 \
log1p.3 log.3 log1pf.3 log.3 log1pl.3 log.3 logf.3 log.3 logl.3 \
log.3 log2.3 log.3 log2f.3 log.3 log2l.3
MLINKS+=lrint.3 llrint.3 lrint.3 llrintf.3 lrint.3 llrintl.3 \
lrint.3 lrintf.3 lrint.3 lrintl.3
MLINKS+=lround.3 llround.3 lround.3 llroundf.3 lround.3 llroundl.3 \

4
lib/msun/Symbol.map
View File

@ -262,4 +262,8 @@ FBSD_1.3 {
ctanh;
ctanhf;
expl;
log10l;
log1pl;
log2l;
logl;
};

737
lib/msun/ld128/s_logl.c Normal file
View File

@ -0,0 +1,737 @@
/*-
* 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 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 */

717
lib/msun/ld80/s_logl.c Normal file
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@ -0,0 +1,717 @@
/*-
* 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 Intel 80-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
#ifdef __i386__
#include <ieeefp.h>
#endif
#include "fpmath.h"
#include "math.h"
#define i386_SSE_GOOD
#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 ~[-5.1736e-22, 5.1738e-22]:
* |log(1 + d)/d - p(d)| < 2**-70.7
*/
static const double
P2 = -0.5,
P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */
P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */
P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */
P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */
P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */
P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */
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) */
double F_lo; /* next 53 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).
*
* We want to share this table between double precision and ld80,
* so the relevant range of dk is the larger one of ld80
* ([-16445, 16383]) and the relevant exactness requirement is
* the stricter one of double precision. The maximum number of
* bits in F_hi(i) that works is very dependent on i but has
* a minimum of 33. We only need about 12 bits in F_hi(i) for
* it to provide enough extra precision in double precision (11
* more than that are required for ld80).
*
* 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, -0x14ee431dae6675.0p-84,
0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84,
0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83,
0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82,
0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82,
0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83,
0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82,
0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91,
0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81,
0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82,
0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83,
0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81,
0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83,
0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85,
0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84,
0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81,
0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82,
0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80,
0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83,
0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82,
0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80,
0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82,
0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81,
0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84,
0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80,
0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81,
0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80,
0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82,
0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81,
0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80,
0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81,
0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81,
0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85,
0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87,
0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81,
0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80,
0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79,
0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79,
0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81,
0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79,
0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81,
0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79,
0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80,
0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81,
0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79,
0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79,
0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81,
0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81,
0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82,
0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80,
0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79,
0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83,
0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79,
0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82,
0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80,
0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79,
0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79,
0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80,
0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79,
0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80,
0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80,
0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79,
0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81,
0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79,
0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81,
0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81,
0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79,
0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80,
0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80,
0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82,
0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80,
0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80,
0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80,
0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80,
0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79,
0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79,
0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80,
0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81,
0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79,
0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84,
0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79,
0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79,
0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79,
0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78,
0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81,
0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79,
0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78,
0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78,
0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79,
0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80,
0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78,
0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78,
0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79,
0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79,
0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78,
0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80,
0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79,
0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79,
0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79,
0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79,
0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78,
0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79,
0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78,
0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79,
0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78,
0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78,
0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78,
0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78,
0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79,
0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79,
0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81,
0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79,
0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78,
0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79,
0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79,
0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78,
0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80,
0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80,
0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79,
0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79,
0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80,
0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79,
0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80,
0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80,
0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81,
0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78,
0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78,
0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81,
};
#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, dk, val_hi, val_lo, z;
uint64_t ix, lx;
int i, k;
uint16_t hx;
EXTRACT_LDBL80_WORDS(hx, lx, 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) == 0)
RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
if (hx != 0)
/* log(neg or [pseudo-]NaN) = qNaN: */
RETURN1(rp, (x - x) / zero);
x *= 0x1.0p65; /* subnormal; scale up x */
/* including pseudo-subnormals */
EXTRACT_LDBL80_WORDS(hx, lx, x);
k = -16383 - 65;
} else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0)
RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
/* log(pseudo-Inf) = qNaN */
/* log(pseudo-NaN) = qNaN */
/* log(unnormal) = qNaN */
#ifndef STRUCT_RETURN
ENTERI();
#endif
k += hx;
ix = lx & 0x7fffffffffffffffULL;
dk = k;
/* Scale x to be in [1, 2). */
SET_LDBL_EXPSIGN(x, 0x3fff);
/* 0 <= i <= INTERVALS: */
#define L2I (64 - LOG2_INTERVALS)
i = (ix + (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, x_lo;
float fx_hi;
/*
* 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.
*/
SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
x_hi = fx_hi;
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.
*/
z = d * d;
val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
(F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * 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, d_lo, dk, f_lo, val_hi, val_lo, z;
long double f_hi, twopminusk;
uint64_t ix, lx;
int i, k;
int16_t ax, hx;
DOPRINT_START(&x);
EXTRACT_LDBL80_WORDS(hx, lx, 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 == 0x8000000000000000ULL)
RETURNP(-1 / zero); /* log1p(-1) = -Inf */
/* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */
RETURNP((x - x) / (x - x));
}
if (ax <= 0x3fbe) { /* |x| < 2**-64 */
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 */
/* log1p(pseudo-Inf) = qNaN */
/* log1p(pseudo-NaN) = qNaN */
/* log1p(unnormal) = qNaN */
} else if (hx < 0x407f) { /* 1 <= x < 2**128 */
f_hi = x;
f_lo = 1;
} else { /* 2**128 <= x < +Inf */
f_hi = x;
f_lo = 0; /* avoid underflow of the P5 term */
}
ENTERI();
x = f_hi + f_lo;
f_lo = (f_hi - x) + f_lo;
EXTRACT_LDBL80_WORDS(hx, lx, x);
k = -16383;
k += hx;
ix = lx & 0x7fffffffffffffffULL;
dk = k;
SET_LDBL_EXPSIGN(x, 0x3fff);
twopminusk = 1;
SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
f_lo *= twopminusk;
i = (ix + (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, x_lo;
float fx_hi;
SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
x_hi = fx_hi;
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;
z = d * d;
val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
(F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * 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);
}
static const double
invln10_hi = 4.3429448190317999e-1, /* 0x1bcb7b1526e000.0p-54 */
invln10_lo = 7.1842412889749798e-14, /* 0x1438ca9aadd558.0p-96 */
invln2_hi = 1.4426950408887933e0, /* 0x171547652b8000.0p-52 */
invln2_lo = 1.7010652264631490e-13; /* 0x17f0bbbe87fed0.0p-95 */
long double
log10l(long double x)
{
struct ld r;
long double hi, lo;
ENTERI();
DOPRINT_START(&x);
k_logl(x, &r);
if (!r.lo_set)
RETURNPI(r.hi);
_2sumF(r.hi, r.lo);
hi = (float)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 hi, lo;
ENTERI();
DOPRINT_START(&x);
k_logl(x, &r);
if (!r.lo_set)
RETURNPI(r.hi);
_2sumF(r.hi, r.lo);
hi = (float)r.hi;
lo = r.lo + (r.hi - hi);
RETURN2PI(invln2_hi * hi,
(invln2_lo + invln2_hi) * lo + invln2_lo * hi);
}
#endif /* STRUCT_RETURN */

41
lib/msun/man/log.3
View File

@ -24,7 +24,7 @@
.\"
.\" $FreeBSD$
.\"
.Dd December 5, 2010
.Dd June 3, 2013
.Dt LOG 3
.Os
.Sh NAME
@ -33,10 +33,13 @@
.Nm logl ,
.Nm log10 ,
.Nm log10f ,
.Nm log10l ,
.Nm log2 ,
.Nm log2f ,
.Nm log2l ,
.Nm log1p ,
.Nm log1pf
.Nm log1pf ,
.Nm log1pl
.Nd logarithm functions
.Sh LIBRARY
.Lb libm
@ -46,43 +49,55 @@
.Fn log "double x"
.Ft float
.Fn logf "float x"
.Ft long double
.Fn logl "long double x"
.Ft double
.Fn log10 "double x"
.Ft float
.Fn log10f "float x"
.Ft long double
.Fn log10l "long double x"
.Ft double
.Fn log2 "double x"
.Ft float
.Fn log2f "float x"
.Ft long double
.Fn log2l "long double x"
.Ft double
.Fn log1p "double x"
.Ft float
.Fn log1pf "float x"
.Ft long double
.Fn log1pl "long double x"
.Sh DESCRIPTION
The
.Fn log
.Fn log ,
.Fn logf ,
and
.Fn logf
.Fn logl
functions compute the natural logarithm of
.Fa x .
.Pp
The
.Fn log10
.Fn log10 ,
.Fn log10f ,
and
.Fn log10f
.Fn log10l
functions compute the logarithm base 10 of
.Fa x ,
while
.Fn log2
.Fn log2 ,
.Fn log2f ,
and
.Fn log2f
.Fn log2l
compute the logarithm base 2 of
.Fa x .
.Pp
The
.Fn log1p
.Fn log1p ,
.Fn log1pf ,
and
.Fn log1pf
.Fn log1pl
functions compute the natural logarithm of
.No "1 + x" .
Computing the natural logarithm as
@ -107,12 +122,16 @@ results in an invalid exception and a return value of \*(Na.
The
.Fn log ,
.Fn logf ,
.Fn logl ,
.Fn log10 ,
.Fn log10f ,
.Fn log10l ,
.Fn log2 ,
.Fn log2f ,
.Fn log2l ,
.Fn log1p ,
.Fn log1pf ,
and
.Fn log1pf
.Fn log1pl
functions conform to
.St -isoC-99 .

6
lib/msun/src/e_log.c
View File

@ -65,6 +65,8 @@ __FBSDID("$FreeBSD$");
* to produce the hexadecimal values shown.
*/
#include <float.h>
#include "math.h"
#include "math_private.h"
@ -139,3 +141,7 @@ __ieee754_log(double x)
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
}
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(log, logl);
#endif

6
lib/msun/src/e_log10.c
View File

@ -22,6 +22,8 @@ __FBSDID("$FreeBSD$");
* in not-quite-routine extra precision.
*/
#include <float.h>
#include "math.h"
#include "math_private.h"
#include "k_log.h"
@ -86,3 +88,7 @@ __ieee754_log10(double x)
return val_lo + val_hi;
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(log10, log10l);
#endif

4
lib/msun/src/e_log2.c
View File

@ -109,3 +109,7 @@ __ieee754_log2(double x)
return val_lo + val_hi;
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(log2, log2l);
#endif

8
lib/msun/src/math.h
View File

@ -418,7 +418,11 @@ int ilogbl(long double) __pure2;
long double ldexpl(long double, int);
long long llrintl(long double);
long long llroundl(long double);
long double log10l(long double);
long double log1pl(long double);
long double log2l(long double);
long double logbl(long double);
long double logl(long double);
long lrintl(long double);
long lroundl(long double);
long double modfl(long double, long double *); /* fundamentally !__pure2 */
@ -464,10 +468,6 @@ long double erfcl(long double);
long double erfl(long double);
long double expm1l(long double);
long double lgammal(long double);
long double log10l(long double);
long double log1pl(long double);
long double log2l(long double);
long double logl(long double);
long double powl(long double, long double);
long double sinhl(long double);
long double tanhl(long double);

292
lib/msun/src/math_private.h
View File

@ -188,6 +188,33 @@ do { \
(d) = sf_u.value; \
} while (0)
/*
* Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
* double.
*/
#define EXTRACT_LDBL80_WORDS(ix0,ix1,d) \
do { \
union IEEEl2bits ew_u; \
ew_u.e = (d); \
(ix0) = ew_u.xbits.expsign; \
(ix1) = ew_u.xbits.man; \
} while (0)
/*
* Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
* long double.
*/
#define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d) \
do { \
union IEEEl2bits ew_u; \
ew_u.e = (d); \
(ix0) = ew_u.xbits.expsign; \
(ix1) = ew_u.xbits.manh; \
(ix2) = ew_u.xbits.manl; \
} while (0)
/* Get expsign as a 16 bit int from a long double. */
#define GET_LDBL_EXPSIGN(i,d) \
@ -197,6 +224,33 @@ do { \
(i) = ge_u.xbits.expsign; \
} while (0)
/*
* Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
* mantissa.
*/
#define INSERT_LDBL80_WORDS(d,ix0,ix1) \
do { \
union IEEEl2bits iw_u; \
iw_u.xbits.expsign = (ix0); \
iw_u.xbits.man = (ix1); \
(d) = iw_u.e; \
} while (0)
/*
* Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
* comprising the mantissa.
*/
#define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2) \
do { \
union IEEEl2bits iw_u; \
iw_u.xbits.expsign = (ix0); \
iw_u.xbits.manh = (ix1); \
iw_u.xbits.manl = (ix2); \
(d) = iw_u.e; \
} while (0)
/* Set expsign of a long double from a 16 bit int. */
#define SET_LDBL_EXPSIGN(d,v) \
@ -260,6 +314,110 @@ do { \
/* Default return statement if hack*_t() is not used. */
#define RETURNF(v) return (v)
/*
* 2sum gives the same result as 2sumF without requiring |a| >= |b| or
* a == 0, but is slower.
*/
#define _2sum(a, b) do { \
__typeof(a) __s, __w; \
\
__w = (a) + (b); \
__s = __w - (a); \
(b) = ((a) - (__w - __s)) + ((b) - __s); \
(a) = __w; \
} while (0)
/*
* 2sumF algorithm.
*
* "Normalize" the terms in the infinite-precision expression a + b for
* the sum of 2 floating point values so that b is as small as possible
* relative to 'a'. (The resulting 'a' is the value of the expression in
* the same precision as 'a' and the resulting b is the rounding error.)
* |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and
* exponent overflow or underflow must not occur. This uses a Theorem of
* Dekker (1971). See Knuth (1981) 4.2.2 Theorem C. The name "TwoSum"
* is apparently due to Skewchuk (1997).
*
* For this to always work, assignment of a + b to 'a' must not retain any
* extra precision in a + b. This is required by C standards but broken
* in many compilers. The brokenness cannot be worked around using
* STRICT_ASSIGN() like we do elsewhere, since the efficiency of this
* algorithm would be destroyed by non-null strict assignments. (The
* compilers are correct to be broken -- the efficiency of all floating
* point code calculations would be destroyed similarly if they forced the
* conversions.)
*
* Fortunately, a case that works well can usually be arranged by building
* any extra precision into the type of 'a' -- 'a' should have type float_t,
* double_t or long double. b's type should be no larger than 'a's type.
* Callers should use these types with scopes as large as possible, to
* reduce their own extra-precision and efficiciency problems. In
* particular, they shouldn't convert back and forth just to call here.
*/
#ifdef DEBUG
#define _2sumF(a, b) do { \
__typeof(a) __w; \
volatile __typeof(a) __ia, __ib, __r, __vw; \
\
__ia = (a); \
__ib = (b); \
assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib)); \
\
__w = (a) + (b); \
(b) = ((a) - __w) + (b); \
(a) = __w; \
\
/* The next 2 assertions are weak if (a) is already long double. */ \
assert((long double)__ia + __ib == (long double)(a) + (b)); \
__vw = __ia + __ib; \
__r = __ia - __vw; \
__r += __ib; \
assert(__vw == (a) && __r == (b)); \
} while (0)
#else /* !DEBUG */
#define _2sumF(a, b) do { \
__typeof(a) __w; \
\
__w = (a) + (b); \
(b) = ((a) - __w) + (b); \
(a) = __w; \
} while (0)
#endif /* DEBUG */
/*
* Set x += c, where x is represented in extra precision as a + b.
* x must be sufficiently normalized and sufficiently larger than c,
* and the result is then sufficiently normalized.
*
* The details of ordering are that |a| must be >= |c| (so that (a, c)
* can be normalized without extra work to swap 'a' with c). The details of
* the normalization are that b must be small relative to the normalized 'a'.
* Normalization of (a, c) makes the normalized c tiny relative to the
* normalized a, so b remains small relative to 'a' in the result. However,
* b need not ever be tiny relative to 'a'. For example, b might be about
* 2**20 times smaller than 'a' to give about 20 extra bits of precision.
* That is usually enough, and adding c (which by normalization is about
* 2**53 times smaller than a) cannot change b significantly. However,
* cancellation of 'a' with c in normalization of (a, c) may reduce 'a'
* significantly relative to b. The caller must ensure that significant
* cancellation doesn't occur, either by having c of the same sign as 'a',
* or by having |c| a few percent smaller than |a|. Pre-normalization of
* (a, b) may help.
*
* This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2
* exercise 19). We gain considerable efficiency by requiring the terms to
* be sufficiently normalized and sufficiently increasing.
*/
#define _3sumF(a, b, c) do { \
__typeof(a) __tmp; \
\
__tmp = (c); \
_2sumF(__tmp, (a)); \
(b) += (a); \
(a) = __tmp; \
} while (0)
/*
* Common routine to process the arguments to nan(), nanf(), and nanl().
*/
@ -370,6 +528,140 @@ irintl(long double x)
#endif /* __GNUCLIKE_ASM */
#ifdef DEBUG
#if defined(__amd64__) || defined(__i386__)
#define breakpoint() asm("int $3")
#else
#include <signal.h>
#define breakpoint() raise(SIGTRAP)
#endif
#endif
/* Write a pari script to test things externally. */
#ifdef DOPRINT
#include <stdio.h>
#ifndef DOPRINT_SWIZZLE
#define DOPRINT_SWIZZLE 0
#endif
#ifdef DOPRINT_LD80
#define DOPRINT_START(xp) do { \
uint64_t __lx; \
uint16_t __hx; \
\
/* Hack to give more-problematic args. */ \
EXTRACT_LDBL80_WORDS(__hx, __lx, *xp); \
__lx ^= DOPRINT_SWIZZLE; \
INSERT_LDBL80_WORDS(*xp, __hx, __lx); \
printf("x = %.21Lg; ", (long double)*xp); \
} while (0)
#define DOPRINT_END1(v) \
printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define DOPRINT_END2(hi, lo) \
printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \
(long double)(hi), (long double)(lo))
#elif defined(DOPRINT_D64)
#define DOPRINT_START(xp) do { \
uint32_t __hx, __lx; \
\
EXTRACT_WORDS(__hx, __lx, *xp); \
__lx ^= DOPRINT_SWIZZLE; \
INSERT_WORDS(*xp, __hx, __lx); \
printf("x = %.21Lg; ", (long double)*xp); \
} while (0)
#define DOPRINT_END1(v) \
printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define DOPRINT_END2(hi, lo) \
printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \
(long double)(hi), (long double)(lo))
#elif defined(DOPRINT_F32)
#define DOPRINT_START(xp) do { \
uint32_t __hx; \
\
GET_FLOAT_WORD(__hx, *xp); \
__hx ^= DOPRINT_SWIZZLE; \
SET_FLOAT_WORD(*xp, __hx); \
printf("x = %.21Lg; ", (long double)*xp); \
} while (0)
#define DOPRINT_END1(v) \
printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define DOPRINT_END2(hi, lo) \
printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \
(long double)(hi), (long double)(lo))
#else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */
#ifndef DOPRINT_SWIZZLE_HIGH
#define DOPRINT_SWIZZLE_HIGH 0
#endif
#define DOPRINT_START(xp) do { \
uint64_t __lx, __llx; \
uint16_t __hx; \
\
EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp); \
__llx ^= DOPRINT_SWIZZLE; \
__lx ^= DOPRINT_SWIZZLE_HIGH; \
INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx); \
printf("x = %.36Lg; ", (long double)*xp); \
} while (0)
#define DOPRINT_END1(v) \
printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define DOPRINT_END2(hi, lo) \
printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n", \
(long double)(hi), (long double)(lo))
#endif /* DOPRINT_LD80 */
#else /* !DOPRINT */
#define DOPRINT_START(xp)
#define DOPRINT_END1(v)
#define DOPRINT_END2(hi, lo)
#endif /* DOPRINT */
#define RETURNP(x) do { \
DOPRINT_END1(x); \
RETURNF(x); \
} while (0)
#define RETURNPI(x) do { \
DOPRINT_END1(x); \
RETURNI(x); \
} while (0)
#define RETURN2P(x, y) do { \
DOPRINT_END2((x), (y)); \
RETURNF((x) + (y)); \
} while (0)
#define RETURN2PI(x, y) do { \
DOPRINT_END2((x), (y)); \
RETURNI((x) + (y)); \
} while (0)
#ifdef STRUCT_RETURN
#define RETURNSP(rp) do { \
if (!(rp)->lo_set) \
RETURNP((rp)->hi); \
RETURN2P((rp)->hi, (rp)->lo); \
} while (0)
#define RETURNSPI(rp) do { \
if (!(rp)->lo_set) \
RETURNPI((rp)->hi); \
RETURN2PI((rp)->hi, (rp)->lo); \
} while (0)
#endif
#define SUM2P(x, y) ({ \
const __typeof (x) __x = (x); \
const __typeof (y) __y = (y); \
\
DOPRINT_END2(__x, __y); \
__x + __y; \
})
/*
* ieee style elementary functions
*

4
lib/msun/src/s_log1p.c
View File

@ -174,3 +174,7 @@ log1p(double x)
if(k==0) return f-(hfsq-s*(hfsq+R)); else
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(log1p, log1pl);
#endif