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log.c
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log.c
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#ifdef sccsid
static char sccsid[] = "@(#)log.c 1.1 92/07/30 SMI";
#endif
/*
* Copyright (c) 1989 by Sun Microsystems, Inc.
*/
/* log(x)
* Table look-up algorithm
* By K.C. Ng, Jan 17, 1989
*
* (a). For x in [31/33,33/31], using a special approximation:
* f = x - 1;
* s = f/(2.0+f); ... here |s| <= 0.03125
* z = s*s;
* return f-s*(f-z*(B1+z*(B2+z*(B3+z*B4))));
*
* (b). For 0.09375 <= x < 23.5
* Use an 8-bit table look-up (3-bit for exponent and 5 bit for
* significand): find a y that match x to 5.5 significand bits,
* then
* log(x) = log(y*(x/y)) = log(y) + log(x/y).
* Here the leading and trailing values of log(y) are obtained from
* a size-512 table. For log(x/y), let s = (x-y)/(x+y), then
* log(x/y) = log((1+s)/(1-s)) = 2s + 2/3 s^3 + 2/5 s^5 +...
* Note that |s|<2**-7=0.0078125. We use an odd s-polynomial
* approximation to compute log(x/y).
*
* (c). Otherwise, get "n", the exponent of x, and then normalize x to
* z in [1,2). Then similar to (b) find a y that matches z to 5.5
* significant bits. Then
* log(x) = n*ln2 + log(y) + log(z/y).
* Here log(y) is obtained from a size-32 table, log(z/y) is
* computed by the same s-polynomial as in (b) (s = (z-y)/(z+y)).
*
* 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.
*
* 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.
*/
extern double SVID_libm_err(),_tbl_log_hi[],_tbl_log_lo[];
static double
zero = 0.0,
one_third = 0.333333333333333333333333,
half = 0.5,
one = 1.0,
two = 2.0,
two52 = 4503599627370496.0,
ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
A1 = 6.666666666674771727490195679289113159045e-0001,
A2 = 3.999999826873793772977541401396094405837e-0001,
A3 = 2.858255498773828156474290815531352737802e-0001,
B1 = 6.666666666666550986298925277570199758720e-0001,
B2 = 4.000000001150663891470403033013455029869e-0001,
B3 = 2.857139299728113781995522584194355262443e-0001,
B4 = 2.226555769734402660003547103954600874552e-0001;
static double TBL[] = {
0.00000000000000000e+00, 3.07716586667536873e-02, 6.06246218164348399e-02,
8.96121586896871380e-02, 1.17783035656383456e-01, 1.45182009844497889e-01,
1.71850256926659228e-01, 1.97825743329919868e-01, 2.23143551314209765e-01,
2.47836163904581269e-01, 2.71933715483641758e-01, 2.95464212893835898e-01,
3.18453731118534589e-01, 3.40926586970593193e-01, 3.62905493689368475e-01,
3.84411698910332056e-01, 4.05465108108164385e-01, 4.26084395310900088e-01,
4.46287102628419530e-01, 4.66089729924599239e-01, 4.85507815781700824e-01,
5.04556010752395312e-01, 5.23248143764547868e-01, 5.41597282432744409e-01,
5.59615787935422659e-01, 5.77315365034823613e-01, 5.94707107746692776e-01,
6.11801541105992941e-01, 6.28608659422374094e-01, 6.45137961373584701e-01,
6.61398482245365016e-01, 6.77398823591806143e-01,
};
double log(x)
double x;
{
double f,s,z,dn;
long *px = (long*)&x;
long *pz = (long*)&z;
int i,j,k,ix,i0,i1,n;
/* get double precision word ordering */
if(*(int*)&one == 0) {i0=1;i1=0;} else {i0=0;i1=1;}
n = 0;
ix = px[i0];
if(ix> 0x3fee0f8e) { /* if x > 31/33 */
if(ix< 0x3ff10842) { /* if x < 33/31 */
f=x-one;
if(((ix+4)&(~7))==0x3ff00000) { /* -2^-19<x-1<2^-18 */
z = f*f;
if(((ix-0x3ff00000)|px[i1])==0) return zero;/*log(1)= +0*/
return f - z*(half-one_third*f);
}
s = f/(two+f); /* |s| <= 0.03125 */
i = (ix-0x3fefe000)|(0x3ff00fff-ix); /* |x-1|<=2**-8 ? */
z = s*s;
if (i>=0) return f-s*(f-z*(B1+z*B2));
return f-s*(f-z*(B1+z*(B2+z*(B3+z*B4))));
}
if(ix>=0x40378000) {
if(ix>=0x7ff00000) return x+x; /* x is +inf or NaN */
goto LARGE_N;
}
goto SMALL_N;
}
if(ix>=0x3fb80000) goto SMALL_N;
if(ix>=0x00100000) goto LARGE_N;
i = ix&0x7fffffff;
if((i|px[i1])==0) return SVID_libm_err(x,x,16); /* log(0.0) = -inf */
if(ix<0) {
if(ix>=0xfff00000) return x-x; /* x is -inf or NaN */
return SVID_libm_err(x,x,17); /* log(x<0) is NaN */
}
/* subnormal x */
x *= two52; n = -52; ix = px[i0];
LARGE_N:
n += ((ix+0x4000)>>20)-0x3ff;
ix = (ix&0x000fffff)|0x3ff00000; /* scale x to [1,2] */
px[i0] = ix;
i = ix + 0x4000;
pz[i0] = i&0xffff8000; pz[i1]=0;
s = (x-z)/(x+z);
j = (i>>15)&0x1f;
z = s*s;
dn = (double) n;
f = dn*ln2lo+s*(two+z*(A1+z*(A2+z*A3)));
f += TBL[j];
return dn*ln2hi+f;
SMALL_N:
pz[i0] = (ix+0x4000)&0xffff8000; pz[i1]=0;
s = (x-z)/(x+z);
j = (ix-0x3fb7c000)>>15; /* combine -(0x4000-0x3fb80000) */
z = s*s;
f = _tbl_log_lo[j]+s*(two+z*(A1+z*(A2+z*A3)));
return _tbl_log_hi[j]+f;
}