Date: Sun, 25 Aug 2013 10:19:10 -0700 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-numerics@FreeBSD.org Subject: Re: (2nd time) tweaks to erff() threshold values Message-ID: <20130825171910.GA60396@troutmask.apl.washington.edu> In-Reply-To: <20130825023029.GA56772@troutmask.apl.washington.edu> References: <20130822213315.GA6708@troutmask.apl.washington.edu> <20130823202257.Q1593@besplex.bde.org> <20130823165744.GA47369@troutmask.apl.washington.edu> <20130824202102.GA54981@troutmask.apl.washington.edu> <20130825023029.GA56772@troutmask.apl.washington.edu>
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On Sat, Aug 24, 2013 at 07:30:29PM -0700, Steve Kargl wrote:
> On Sat, Aug 24, 2013 at 01:21:02PM -0700, Steve Kargl wrote:
> > On Fri, Aug 23, 2013 at 09:57:44AM -0700, Steve Kargl wrote:
> > > On Fri, Aug 23, 2013 at 09:12:33PM +1000, Bruce Evans wrote:
> > >
> > >> The whole erf implementation is probably not the best way for
> > >> float precision, but is good for understanding higher precisions.
> > >
> > > I'm fairly certain that the implementation of erff() is not very
> > > efficient. The polynomials, used in the rational approximations,
> > > are the same order as those used in the double precision approximation.
> > > I'll check the polys when update my P(x)/Q(x) remes algorithm for
> > > erfl and erfcl.
> >
> > I seem to be right (although I haven't iterated on the P's and Q's).
> >
>
> Now, with pretty testing and (perhaps) better coefficients:
These values where for the interval [0,0.84375].
>
> | usec/call | Max ULP | X Max ULP
> ---------+-----------+---------+---------------------------------
> new erfc | 0.02757 | 0.65947 | 8.19824636e-01, 0x1.a3c00ep-1
> old erfc | 0.03348 | 0.68218 | 8.43257010e-01, 0x1.afbf62p-1
> ---------+-----------+---------+---------------------------------
> new erf | 0.02302 | 0.62437 | 1.04175471e-38, 0x1.c5bf88p-127
> old erf | 0.03028 | 0.62437 | 1.04175471e-38, 0x1.c5bf88p-127
New interval [0.84375,1.25].
| usecs/call| Max ULP | X Max ULP
---------+-----------+---------+------------------------------
Old erfc | 0.03469 | 0.95158 | 1.24999416e+00, 0x1.3fff9ep+0
New erfc | 0.02781 | 0.76497 | 1.24971473e+00, 0x1.3fed4ep+0
---------+-----------+---------+------------------------------
Old erf | 0.03404 | 0.55617 | 1.24996567e+00, 0x1.3ffdcp+0
New erf | 0.02774 | 0.54213 | 8.52452755e-01, 0x1.b474bp-1
Note that s_erf.c claims that in this interval, a taylor series about
erf(1) is used and suggests that the constant erx = 8.4506291151e-01
is erf(1). That's bogus as erf(1) = 8.42700779e-01F (in float).
Note**2, I did not record a domain and range as my routine that
generates plots seems to give a really messed up result for
the range [-0.08, 0.06]. However, the error estimate from
solving the Remes matrix equation is err = 0x1.39c84809b7ed2p-38.
New diff:
Index: src/s_erff.c
===================================================================
--- src/s_erff.c (revision 1361)
+++ src/s_erff.c (working copy)
@@ -25,38 +25,34 @@ half= 5.0000000000e-01, /* 0x3F000000 *
one = 1.0000000000e+00, /* 0x3F800000 */
two = 2.0000000000e+00, /* 0x40000000 */
/* c = (subfloat)0.84506291151 */
-erx = 8.4506291151e-01, /* 0x3f58560b */
+erx = 8.42700779e-01F, /* 0x1.af767ap-1 */
+
/*
* Coefficients for approximation to erf on [0,0.84375]
*/
efx = 1.2837916613e-01, /* 0x3e0375d4 */
efx8= 1.0270333290e+00, /* 0x3f8375d4 */
-pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
-pp1 = -3.2504209876e-01, /* 0xbea66beb */
-pp2 = -2.8481749818e-02, /* 0xbce9528f */
-pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
-pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
-qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
-qq2 = 6.5022252500e-02, /* 0x3d852a63 */
-qq3 = 5.0813062117e-03, /* 0x3ba68116 */
-qq4 = 1.3249473704e-04, /* 0x390aee49 */
-qq5 = -3.9602282413e-06, /* 0xb684e21a */
+/*
+ * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
+ * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
+ */
+pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */
+pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */
+pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */
+qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */
+qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */
+qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */
/*
* Coefficients for approximation to erf in [0.84375,1.25]
*/
-pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
-pa1 = 4.1485610604e-01, /* 0x3ed46805 */
-pa2 = -3.7220788002e-01, /* 0xbebe9208 */
-pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
-pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
-pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
-pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
-qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
-qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
-qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
-qa4 = 1.2617121637e-01, /* 0x3e013307 */
-qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
-qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
+pa0 = 1.35131621e-08F, /* 0x1.d04f08p-27 */
+pa1 = 4.15107518e-01F, /* 0x1.a911f2p-2 */
+pa2 = -1.63339108e-01F, /* -0x1.4e84bcp-3 */
+pa3 = 1.12098485e-01F, /* 0x1.cb27c8p-4 */
+qa1 = 6.06513679e-01F, /* 0x1.3688f6p-1 */
+qa2 = 5.43227255e-01F, /* 0x1.1621e2p-1 */
+qa3 = 1.74396917e-01F, /* 0x1.652a36p-3 */
+qa4 = 5.88681065e-02F, /* 0x1.e23f5ep-5 */
/*
* Coefficients for approximation to erfc in [1.25,1/0.35]
*/
@@ -114,15 +110,15 @@ erff(float x)
return x + efx*x;
}
z = x*x;
- r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
- s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+ r = pp0+z*(pp1+z*pp2);
+ s = one+z*(qq1+z*(qq2+z*qq3));
y = r/s;
return x + x*y;
}
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
- P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
- Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+ P = pa0+s*(pa1+s*(pa2+s*pa3));
+ Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
}
if (ix >= 0x40c00000) { /* inf>|x|>=6 */
@@ -163,8 +159,8 @@ erfcf(float x)
if(ix < 0x23800000) /* |x|<2**-56 */
return one-x;
z = x*x;
- r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
- s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+ r = pp0+z*(pp1+z*pp2);
+ s = one+z*(qq1+z*(qq2+z*qq3));
y = r/s;
if(hx < 0x3e800000) { /* x<1/4 */
return one-(x+x*y);
@@ -176,8 +172,8 @@ erfcf(float x)
}
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
- P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
- Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+ P = pa0+s*(pa1+s*(pa2+s*pa3));
+ Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
if(hx>=0) {
z = one-erx; return z - P/Q;
} else {
--
Steve
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