* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* 1. Redistributions of source code must retain the above copyright
* notice, 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.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``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 REGENTS OR CONTRIBUTORS 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
static char sccsid
[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
/* Table-driven natural logarithm.
* This code was derived, with minor modifications, from:
* Peter Tang, "Table-Driven Implementation of the
* Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
* Math Software, vol 16. no 4, pp 378-400, Dec 1990).
* Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
* where F = j/128 for j an integer in [0, 128].
* log(2^m) = log2_hi*m + log2_tail*m
* since m is an integer, the dominant term is exact.
* m has at most 10 digits (for subnormal numbers),
* and log2_hi has 11 trailing zero bits.
* log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
* logF_hi[] + 512 is exact.
* log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
* the leading term is calculated to extra precision in two
* parts, the larger of which adds exactly to the dominant
* 1. when m, j are non-zero (m | j), use absolute
* precision for the leading term.
* 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
* In this case, use a relative precision of 24 bits.
* (This is done differently in the original paper)
* 0 return signalling -Inf
* neg return signalling NaN
#if defined(vax) || defined(tahoe)
#define TRUNC(x) x = (double) (float) (x)
#define endian (((*(int *) &one)) ? 1 : 0)
#define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
* Used for generation of extend precision logarithms.
* The constant 35184372088832 is 2^45, so the divide is exact.
* It ensures correct reading of logF_head, even for inaccurate
* decimal-to-binary conversion routines. (Everybody gets the
* right answer for integers less than 2^53.)
* Values for log(F) were generated using error < 10^-57 absolute
* with the bc -l package.
static double A1
= .08333333333333178827;
static double A2
= .01250000000377174923;
static double A3
= .002232139987919447809;
static double A4
= .0004348877777076145742;
static double logF_head
[N
+1] = {
static double logF_tail
[N
+1] = {
-.00000000000000543229938420049,
.00000000000000172745674997061,
-.00000000000001323017818229233,
-.00000000000001154527628289872,
-.00000000000000466529469958300,
.00000000000005148849572685810,
-.00000000000002532168943117445,
-.00000000000005213620639136504,
-.00000000000001819506003016881,
.00000000000006329065958724544,
.00000000000008614512936087814,
-.00000000000007355770219435028,
.00000000000009638067658552277,
.00000000000007598636597194141,
.00000000000002579999128306990,
-.00000000000004654729747598444,
-.00000000000007556920687451336,
.00000000000010195735223708472,
-.00000000000017319034406422306,
-.00000000000007718001336828098,
.00000000000010980754099855238,
-.00000000000002047235780046195,
-.00000000000008372091099235912,
.00000000000014088127937111135,
.00000000000012869017157588257,
.00000000000017788850778198106,
.00000000000006440856150696891,
.00000000000016132822667240822,
-.00000000000007540916511956188,
-.00000000000000036507188831790,
.00000000000009120937249914984,
.00000000000018567570959796010,
-.00000000000003149265065191483,
-.00000000000009309459495196889,
.00000000000017914338601329117,
-.00000000000001302979717330866,
.00000000000023097385217586939,
.00000000000023999540484211737,
.00000000000015393776174455408,
-.00000000000036870428315837678,
.00000000000036920375082080089,
-.00000000000009383417223663699,
.00000000000009433398189512690,
.00000000000041481318704258568,
-.00000000000003792316480209314,
.00000000000008403156304792424,
-.00000000000034262934348285429,
.00000000000043712191957429145,
-.00000000000010475750058776541,
-.00000000000011118671389559323,
.00000000000037549577257259853,
.00000000000013912841212197565,
.00000000000010775743037572640,
.00000000000029391859187648000,
-.00000000000042790509060060774,
.00000000000022774076114039555,
.00000000000010849569622967912,
-.00000000000023073801945705758,
.00000000000015761203773969435,
.00000000000003345710269544082,
-.00000000000041525158063436123,
.00000000000032655698896907146,
-.00000000000044704265010452446,
.00000000000034527647952039772,
-.00000000000007048962392109746,
.00000000000011776978751369214,
-.00000000000010774341461609578,
.00000000000021863343293215910,
.00000000000024132639491333131,
.00000000000039057462209830700,
-.00000000000026570679203560751,
.00000000000037135141919592021,
-.00000000000017166921336082431,
-.00000000000028658285157914353,
-.00000000000023812542263446809,
.00000000000006576659768580062,
-.00000000000028210143846181267,
.00000000000010701931762114254,
.00000000000018119346366441110,
.00000000000009840465278232627,
-.00000000000033149150282752542,
-.00000000000018302857356041668,
-.00000000000016207400156744949,
.00000000000048303314949553201,
-.00000000000071560553172382115,
.00000000000088821239518571855,
-.00000000000030900580513238244,
-.00000000000061076551972851496,
.00000000000035659969663347830,
.00000000000035782396591276383,
-.00000000000046226087001544578,
.00000000000062279762917225156,
.00000000000072838947272065741,
.00000000000026809646615211673,
-.00000000000010960825046059278,
.00000000000002311949383800537,
-.00000000000058469058005299247,
-.00000000000002103748251144494,
-.00000000000023323182945587408,
-.00000000000042333694288141916,
-.00000000000043933937969737844,
.00000000000041341647073835565,
.00000000000006841763641591466,
.00000000000047585534004430641,
.00000000000083679678674757695,
-.00000000000085763734646658640,
.00000000000021913281229340092,
-.00000000000062242842536431148,
-.00000000000010983594325438430,
.00000000000065310431377633651,
-.00000000000047580199021710769,
-.00000000000037854251265457040,
.00000000000040939233218678664,
.00000000000087424383914858291,
.00000000000025218188456842882,
-.00000000000003608131360422557,
-.00000000000050518555924280902,
.00000000000078699403323355317,
-.00000000000067020876961949060,
.00000000000016108575753932458,
.00000000000058527188436251509,
-.00000000000035246757297904791,
-.00000000000018372084495629058,
.00000000000088606689813494916,
.00000000000066486268071468700,
.00000000000063831615170646519,
.00000000000025144230728376072,
-.00000000000017239444525614834
double F
, f
, g
, q
, u
, u2
, v
, zero
= 0.0, one
= 1.0;
/* Catch special cases */
if (_IEEE
&& x
== zero
) /* log(0) = -Inf */
else if (_IEEE
) /* log(neg) = NaN */
else if (x
== zero
) /* NOT REACHED IF _IEEE */
return (infnan(-ERANGE
));
if (_IEEE
) /* x = NaN, Inf */
/* Argument reduction: 1 <= g < 2; x/2^m = g; */
/* y = F*(1 + f/F) for |f| <= 2^-8 */
if (_IEEE
&& m
== -1022) {
F
= (1.0/N
) * j
+ 1; /* F*128 is an integer in [128, 512] */
/* Approximate expansion for log(1+f/F) ~= u + q */
q
= u
*v
*(A1
+ v
*(A2
+ v
*(A3
+ v
*A4
)));
/* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
* u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
* It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
/* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
u2
= (2.0*(f
- F
*u1
) - u1
*f
) * g
;
/* u1 + u2 = 2f/(2F+f) to extra precision. */
/* log(x) = log(2^m*F*(1+f/F)) = */
/* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
u1
+= m
*logF_head
[N
] + logF_head
[j
]; /* exact */
u2
= (u2
+ logF_tail
[j
]) + q
; /* tiny */
* Extra precision variant, returning struct {double a, b;};
* log(x) = a+b to 63 bits, with a is rounded to 26 bits.
double F
, f
, g
, q
, u
, v
, u2
, one
= 1.0;
/* Argument reduction: 1 <= g < 2; x/2^m = g; */
/* y = F*(1 + f/F) for |f| <= 2^-8 */
if (_IEEE
&& m
== -1022) {
q
= u
*v
*(A1
+ v
*(A2
+ v
*(A3
+ v
*A4
)));
u2
= (2.0*(f
- F
*u1
) - u1
*f
) * g
;
u1
+= m
*logF_head
[N
] + logF_head
[j
];
u2
+= logF_tail
[j
]; u2
+= q
;
r
.a
= u1
+ u2
; /* Only difference is here */