This commit was manufactured by cvs2svn to create tag 'FreeBSD-release/1.0'.

[unix-history] / lib / libm / common_source / log.c

diff --git a/lib/libm/common_source/log.c b/lib/libm/common_source/log.c
index d1dec5b..f565dcd 100644 (file)
--- a/lib/libm/common_source/log.c
+++ b/lib/libm/common_source/log.c
@@ -1,5 +1,5 @@
 /*
 /*

- * Copyright (c) 1985 Regents of the University of California.
+ * Copyright (c) 1992 Regents of the University of California.

  * All rights reserved.
  *
  * Redistribution and use in source and binary forms, with or without
  * All rights reserved.
  *
  * Redistribution and use in source and binary forms, with or without


@@ -32,132 +32,457 @@
  */
 
 #ifndef lint
  */
 
 #ifndef lint

-static char sccsid[] = "@(#)log.c      5.6 (Berkeley) 10/9/90";
+static char sccsid[] = "@(#)log.c      5.10 (Berkeley) 1/10/93";

 #endif /* not lint */
 
 #endif /* not lint */
 

-/* LOG(X)
- * RETURN THE LOGARITHM OF x 
- * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
- * CODED IN C BY K.C. NG, 1/19/85;
- * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85.
- *
- * Required system supported functions:
- *     scalb(x,n)
- *     copysign(x,y)
- *     logb(x) 
- *     finite(x)
- *
- * Required kernel function:
- *     log__L(z) 
+#include <math.h>
+#include <errno.h>
+
+#include "mathimpl.h"
+
+/* Table-driven natural logarithm.

  *
  *

- * Method :
- *     1. Argument Reduction: find k and f such that 
- *                     x = 2^k * (1+f), 
- *        where  sqrt(2)/2 < 1+f < sqrt(2) .
+ * 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).

  *
  *

- *     2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
- *              = 2s + 2/3 s**3 + 2/5 s**5 + .....,
- *        log(1+f) is computed by
+ * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
+ * where F = j/128 for j an integer in [0, 128].

  *
  *

- *                     log(1+f) = 2s + s*log__L(s*s)
- *        where
- *             log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
+ * 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.

  *
  *

- *        See log__L() for the values of the coefficients.
+ * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
+ * logF_hi[] + 512 is exact.

  *
  *

- *     3. Finally,  log(x) = k*ln2 + log(1+f).  (Here n*ln2 will be stored
- *        in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
- *        since the last 20 bits of ln2hi is 0.)
+ * 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
+ * m and F terms.
+ * There are two cases:
+ *     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)

  *
  * Special cases:
  *
  * Special cases:

- *     log(x) is NaN with signal if x < 0 (including -INF) ; 
- *     log(+INF) is +INF; log(0) is -INF with signal;
- *     log(NaN) is that NaN with no signal.
- *
- * Accuracy:
- *     log(x) returns the exact log(x) nearly rounded. In a test run with
- *     1,536,000 random arguments on a VAX, the maximum observed error was
- *     .826 ulps (units in the last place).
- *
- * Constants:
- * The hexadecimal values are the intended ones for the following constants.
- * The decimal values may be used, provided that the compiler will convert
- * from decimal to binary accurately enough to produce the hexadecimal values
- * shown.
- */
+ *     0       return signalling -Inf
+ *     neg     return signalling NaN
+ *     +Inf    return +Inf
+*/

 
 

-#include <errno.h>
-#include "mathimpl.h"
+#if defined(vax) || defined(tahoe)
+#define _IEEE          0
+#define TRUNC(x)       x = (double) (float) (x)
+#else
+#define _IEEE          1
+#define endian         (((*(int *) &one)) ? 1 : 0)
+#define TRUNC(x)       *(((int *) &x) + endian) &= 0xf8000000
+#define infnan(x)      0.0
+#endif

 
 

-vc(ln2hi, 6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
-vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
-vc(sqrt2, 1.4142135623730950622E0   ,04f3,40b5,de65,33f9,   1, .B504F333F9DE65)
+#define N 128

 
 

-ic(ln2hi, 6.9314718036912381649E-1,   -1, 1.62E42FEE00000)
-ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
-ic(sqrt2, 1.4142135623730951455E0,     0, 1.6A09E667F3BCD)
+/* 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.
+*/
+double A1 =      .08333333333333178827;
+double A2 =      .01250000000377174923;
+double A3 =     .002232139987919447809;
+double A4 =    .0004348877777076145742;

 
 

-#ifdef vccast
-#define        ln2hi   vccast(ln2hi)
-#define        ln2lo   vccast(ln2lo)
-#define        sqrt2   vccast(sqrt2)
-#endif
+double logF_head[N+1] = {
+       0.,
+       .007782140442060381246,
+       .015504186535963526694,
+       .023167059281547608406,
+       .030771658666765233647,
+       .038318864302141264488,
+       .045809536031242714670,
+       .053244514518837604555,
+       .060624621816486978786,
+       .067950661908525944454,
+       .075223421237524235039,
+       .082443669210988446138,
+       .089612158689760690322,
+       .096729626458454731618,
+       .103796793681567578460,
+       .110814366340264314203,
+       .117783035656430001836,
+       .124703478501032805070,
+       .131576357788617315236,
+       .138402322859292326029,
+       .145182009844575077295,
+       .151916042025732167530,
+       .158605030176659056451,
+       .165249572895390883786,
+       .171850256926518341060,
+       .178407657472689606947,
+       .184922338493834104156,
+       .191394852999565046047,
+       .197825743329758552135,
+       .204215541428766300668,
+       .210564769107350002741,
+       .216873938300523150246,
+       .223143551314024080056,
+       .229374101064877322642,
+       .235566071312860003672,
+       .241719936886966024758,
+       .247836163904594286577,
+       .253915209980732470285,
+       .259957524436686071567,
+       .265963548496984003577,
+       .271933715484010463114,
+       .277868451003087102435,
+       .283768173130738432519,
+       .289633292582948342896,
+       .295464212893421063199,
+       .301261330578199704177,
+       .307025035294827830512,
+       .312755710004239517729,
+       .318453731118097493890,
+       .324119468654316733591,
+       .329753286372579168528,
+       .335355541920762334484,
+       .340926586970454081892,
+       .346466767346100823488,
+       .351976423156884266063,
+       .357455888922231679316,
+       .362905493689140712376,
+       .368325561158599157352,
+       .373716409793814818840,
+       .379078352934811846353,
+       .384411698910298582632,
+       .389716751140440464951,
+       .394993808240542421117,
+       .400243164127459749579,
+       .405465108107819105498,
+       .410659924985338875558,
+       .415827895143593195825,
+       .420969294644237379543,
+       .426084395310681429691,
+       .431173464818130014464,
+       .436236766774527495726,
+       .441274560805140936281,
+       .446287102628048160113,
+       .451274644139630254358,
+       .456237433481874177232,
+       .461175715122408291790,
+       .466089729924533457960,
+       .470979715219073113985,
+       .475845904869856894947,
+       .480688529345570714212,
+       .485507815781602403149,
+       .490303988045525329653,
+       .495077266798034543171,
+       .499827869556611403822,
+       .504556010751912253908,
+       .509261901790523552335,
+       .513945751101346104405,
+       .518607764208354637958,
+       .523248143765158602036,
+       .527867089620485785417,
+       .532464798869114019908,
+       .537041465897345915436,
+       .541597282432121573947,
+       .546132437597407260909,
+       .550647117952394182793,
+       .555141507540611200965,
+       .559615787935399566777,
+       .564070138285387656651,
+       .568504735352689749561,
+       .572919753562018740922,
+       .577315365035246941260,
+       .581691739635061821900,
+       .586049045003164792433,
+       .590387446602107957005,
+       .594707107746216934174,
+       .599008189645246602594,
+       .603290851438941899687,
+       .607555250224322662688,
+       .611801541106615331955,
+       .616029877215623855590,
+       .620240409751204424537,
+       .624433288012369303032,
+       .628608659422752680256,
+       .632766669570628437213,
+       .636907462236194987781,
+       .641031179420679109171,
+       .645137961373620782978,
+       .649227946625615004450,
+       .653301272011958644725,
+       .657358072709030238911,
+       .661398482245203922502,
+       .665422632544505177065,
+       .669430653942981734871,
+       .673422675212350441142,
+       .677398823590920073911,
+       .681359224807238206267,
+       .685304003098281100392,
+       .689233281238557538017,
+       .693147180560117703862
+};

 
 

+double logF_tail[N+1] = {
+       0.,
+       -.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 log(x)
-double x;
+double
+#ifdef _ANSI_SOURCE
+log(double x)
+#else
+log(x) double x;
+#endif

 {
 {

-       const static double zero=0.0, negone= -1.0, half=1.0/2.0;
-       double s,z,t;
-       int k,n;
-
-#if !defined(vax)&&!defined(tahoe)
-       if(x!=x) return(x);     /* x is NaN */
-#endif /* !defined(vax)&&!defined(tahoe) */
-       if(finite(x)) {
-          if( x > zero ) {
-
-          /* argument reduction */
-             k=logb(x);   x=scalb(x,-k);
-             if(k == -1022) /* subnormal no. */
-                  {n=logb(x); x=scalb(x,-n); k+=n;} 
-             if(x >= sqrt2 ) {k += 1; x *= half;}
-             x += negone ;
-
-          /* compute log(1+x)  */
-              s=x/(2+x); t=x*x*half;
-             z=k*ln2lo+s*(t+log__L(s*s));
-             x += (z - t) ;
-
-             return(k*ln2hi+x);
-          }
-       /* end of if (x > zero) */
-
-          else {
-#if defined(vax)||defined(tahoe)
-               if ( x == zero )
-                   return (infnan(-ERANGE));   /* -INF */
+       int m, j;
+       double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
+       double logb(), ldexp();
+       volatile double u1;
+
+       /* Catch special cases */
+       if (x <= 0)
+               if (_IEEE && x == zero) /* log(0) = -Inf */
+                       return (-one/zero);
+               else if (_IEEE)         /* log(neg) = NaN */
+                       return (zero/zero);
+               else if (x == zero)     /* NOT REACHED IF _IEEE */
+                       return (infnan(-ERANGE));
+               else
+                       return (infnan(EDOM));
+       else if (!finite(x))
+               if (_IEEE)              /* x = NaN, Inf */
+                       return (x+x);

                else
                else

-                   return (infnan(EDOM));      /* NaN */
-#else  /* defined(vax)||defined(tahoe) */
-               /* zero argument, return -INF with signal */
-               if ( x == zero )
-                   return( negone/zero );
-
-               /* negative argument, return NaN with signal */
-               else 
-                   return ( zero / zero );
-#endif /* defined(vax)||defined(tahoe) */
-           }
+                       return (infnan(ERANGE));
+       
+       /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
+       /* y = F*(1 + f/F) for |f| <= 2^-8              */
+
+       m = logb(x);
+       g = ldexp(x, -m);
+       if (_IEEE && m == -1022) {
+               j = logb(g), m += j;
+               g = ldexp(g, -j);
+       }
+       j = N*(g-1) + .5;
+       F = (1.0/N) * j + 1;    /* F*128 is an integer in [128, 512] */
+       f = g - F;
+
+       /* Approximate expansion for log(1+f/F) ~= u + q */
+       g = 1/(2*F+f);
+       u = 2*f*g;
+       v = u*u;
+       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
+    */
+       if (m | j)
+               u1 = u + 513, u1 -= 513;
+
+    /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
+     *                 u1 = u to 24 bits.
+    */
+       else
+               u1 = u, TRUNC(u1);
+       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);    */
+       /* (exact) + (tiny)                                             */
+
+       u1 += m*logF_head[N] + logF_head[j];            /* exact */
+       u2 = (u2 + logF_tail[j]) + q;                   /* tiny */
+       u2 += logF_tail[N]*m;
+       return (u1 + u2);
+}
+
+/*
+ * Extra precision variant, returning struct {double a, b;};
+ * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
+ */
+struct Double
+#ifdef _ANSI_SOURCE
+log__D(double x)
+#else
+log__D(x) double x;
+#endif
+{
+       int m, j;
+       double F, f, g, q, u, v, u2, one = 1.0;
+       double logb(), ldexp();
+       volatile double u1;
+       struct Double r;
+
+       /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
+       /* y = F*(1 + f/F) for |f| <= 2^-8              */
+
+       m = logb(x);
+       g = ldexp(x, -m);
+       if (_IEEE && m == -1022) {
+               j = logb(g), m += j;
+               g = ldexp(g, -j);

        }
        }

-    /* end of if (finite(x)) */
-    /* NOTREACHED if defined(vax)||defined(tahoe) */
+       j = N*(g-1) + .5;
+       F = (1.0/N) * j + 1;
+       f = g - F;

 
 

-    /* log(-INF) is NaN with signal */
-       else if (x<0) 
-           return(zero/zero);      
+       g = 1/(2*F+f);
+       u = 2*f*g;
+       v = u*u;
+       q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
+       if (m | j)
+               u1 = u + 513, u1 -= 513;
+       else
+               u1 = u, TRUNC(u1);
+       u2 = (2.0*(f - F*u1) - u1*f) * g;

 
 

-    /* log(+INF) is +INF */
-       else return(x);      
+       u1 += m*logF_head[N] + logF_head[j];

 
 

+       u2 +=  logF_tail[j]; u2 += q;
+       u2 += logF_tail[N]*m;
+       r.a = u1 + u2;                  /* Only difference is here */
+       TRUNC(r.a);
+       r.b = (u1 - r.a) + u2;
+       return (r);

 }
 }