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For log1pf(), fixed the approximations to sqrt(2), sqrt(2)-1 and

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sqrt(2)/2-1.  For log1p(), fixed the approximation to sqrt(2)/2-1.

The end result is to fix an error of 1.293 ulps in
    log1pf(0.41421395540 (hex 0x3ed413da))
and an error of 1.783 ulps in
    log1p(-0.292893409729003961761) (hex 0x12bec4 00000001)).
The former was the only error of > 1 ulp for log1pf() and the latter
is the only such error that I know of for log1p().

The approximations don't need to be very accurate, but the last 2 need
to be related to the first and be rounded up a little (even more than
1 ulp for sqrt(2)/2-1) for the following implementation-detail reason:
when the arg (x) is not between (the approximations to) sqrt(2)/2-1
and sqrt(2)-1, we commit to using a correction term, but we only
actually use it if 1+x is between sqrt(2)/2 and sqrt(2) according to
the first approximation. Thus we must ensure that
!(sqrt(2)/2-1 < x < sqrt(2)-1) implies !(sqrt(2)/2 < x+1 < sqrt(2)),
where all the sqrt(2)'s are really slightly different approximations
to sqrt(2) and some of the "<"'s are really "<="'s.  This was not done.

In log1pf(), the last 2 approximations were rounded up by about 6 ulps
more than needed relative to a good approximation to sqrt(2), but the
actual approximation to sqrt(2) was off by 3 ulps.  The approximation
to sqrt(2)-1 ended up being 4 ulps too small, so the algoritm was
broken in 4 cases.  The result happened to be broken in 1 case.  This
is fixed by using a natural approximation to sqrt(2) and derived
approximations for the others.

In logf(), all the approximations made sense, but the approximation
to sqrt(2)/2-1 was 2 ulps too small (a tiny amount, since we compare
with a granularity of 2**32 ulps), so the algorithm was broken in 2
cases.  The result was broken in 1 case.  This is fixed by rounding
up the approximation to sqrt(2)/2-1 by 2**32 ulps, so 2**32 cases are
now handled a little differently (still correctly according to my
assertion that the approximations don't need to be very accurate, but
this has not been checked).
This commit is contained in:
Bruce Evans 2005-12-04 12:30:44 +00:00
parent 69a7db097e
commit d48ea9753c
Notes: svn2git 2020-12-20 02:59:44 +00:00 
svn path=/head/; revision=153086
2 changed files with 22 additions and 8 deletions
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15
lib/msun/src/s_log1p.c
View file

@ -106,7 +106,7 @@ log1p(double x)
ax = hx&0x7fffffff;
k = 1;
if (hx < 0x3FDA827A) { /* x < 0.41422 */
if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
if(ax>=0x3ff00000) { /* x <= -1.0 */
if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
@ -118,8 +118,8 @@ log1p(double x)
else
return x - x*x*0.5;
}
if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
}
if (hx >= 0x7ff00000) return x+x;
if(k!=0) {
@ -136,7 +136,14 @@ log1p(double x)
c = 0;
}
hu &= 0x000fffff;
if(hu<0x6a09e) {
/*
* The approximation to sqrt(2) used in thresholds is not
* critical. However, the ones used above must give less
* strict bounds than the one here so that the k==0 case is
* never reached from here, since here we have committed to
* using the correction term but don't use it if k==0.
*/
if(hu<0x6a09e) { /* u ~< sqrt(2) */
SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
} else {
k += 1;

15
lib/msun/src/s_log1pf.c
View file

@ -44,7 +44,7 @@ log1pf(float x)
ax = hx&0x7fffffff;
k = 1;
if (hx < 0x3ed413d7) { /* x < 0.41422 */
if (hx < 0x3ed413d0) { /* 1+x < sqrt(2)+ */
if(ax>=0x3f800000) { /* x <= -1.0 */
if(x==(float)-1.0) return -two25/zero; /* log1p(-1)=+inf */
else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
@ -56,8 +56,8 @@ log1pf(float x)
else
return x - x*x*(float)0.5;
}
if(hx>0||hx<=((int32_t)0xbe95f61f)) {
k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
if(hx>0||hx<=((int32_t)0xbe95f619)) {
k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
}
if (hx >= 0x7f800000) return x+x;
if(k!=0) {
@ -75,7 +75,14 @@ log1pf(float x)
c = 0;
}
hu &= 0x007fffff;
if(hu<0x3504f7) {
/*
* The approximation to sqrt(2) used in thresholds is not
* critical. However, the ones used above must give less
* strict bounds than the one here so that the k==0 case is
* never reached from here, since here we have committed to
* using the correction term but don't use it if k==0.
*/
if(hu<0x3504f4) { /* u < sqrt(2) */
SET_FLOAT_WORD(u,hu|0x3f800000);/* normalize u */
} else {
k += 1;