* s_erf.c:

  . Use integer literal constants instead of double literal constants.

* s_erff.c:
  . Use integer literal constants instead of casting double literal
    constants to float.
  . Update the threshold values from those carried over from erf() to
    values appropriate for float.
  . New sets of polynomial coefficients for the rational approximations.
    These coefficients have little, but positive, effect on the maximum
    error in ULP in the four intervals, but do improve the overall
    speed of execution.
  . Remove redundant GET_FLOAT_WORD(ix,x) as hx already contained the
    contents that is packed into ix.
  . Update the mask that is used to zero-out lower-order bits in x in
    the intervals [1.25, 2.857143] and [2.857143, 12].  In tests on
    amd64, this change improves the maximum error in ULP from 6.27739
    and 63.8095 to 3.16774 and 2.92095 on these intervals for erffc().

Reviewed by:	bde

2 files changed, 71 insertions, 99 deletions

diff --git a/lib/msun/src/s_erf.c b/lib/msun/src/s_erf.c
index 0886e5e..a4e91a3 100644
--- a/lib/msun/src/s_erf.c
+++ b/lib/msun/src/s_erf.c
@@ -201,7 +201,7 @@ erf(double x)
 	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
 	    if(ix < 0x3e300000) { 	/* |x|<2**-28 */
 	        if (ix < 0x00800000)
-		    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
+		    return (8*x+efx8*x)/8;  /* avoid spurious underflow */
 		return x + efx*x;
 	    }
 	    z = x*x;
diff --git a/lib/msun/src/s_erff.c b/lib/msun/src/s_erff.c
index a44e135..b97ca1d 100644
--- a/lib/msun/src/s_erff.c
+++ b/lib/msun/src/s_erff.c
@@ -24,75 +24,59 @@ tiny	    = 1e-30,
 half=  5.0000000000e-01, /* 0x3F000000 */
 one =  1.0000000000e+00, /* 0x3F800000 */
 two =  2.0000000000e+00, /* 0x40000000 */
-	/* c = (subfloat)0.84506291151 */
-erx =  8.4506291151e-01, /* 0x3f58560b */
 /*
- * Coefficients for approximation to  erf on [0,0.84375]
+ * 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 */
 /*
- * Coefficients for approximation to  erf  in [0.84375,1.25]
+ *  Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
+ *  |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
  */
-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 */
+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  erfc in [1.25,1/0.35]
+ * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
+ * |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
  */
-ra0  = -9.8649440333e-03, /* 0xbc21a093 */
-ra1  = -6.9385856390e-01, /* 0xbf31a0b7 */
-ra2  = -1.0558626175e+01, /* 0xc128f022 */
-ra3  = -6.2375331879e+01, /* 0xc2798057 */
-ra4  = -1.6239666748e+02, /* 0xc322658c */
-ra5  = -1.8460508728e+02, /* 0xc3389ae7 */
-ra6  = -8.1287437439e+01, /* 0xc2a2932b */
-ra7  = -9.8143291473e+00, /* 0xc11d077e */
-sa1  =  1.9651271820e+01, /* 0x419d35ce */
-sa2  =  1.3765776062e+02, /* 0x4309a863 */
-sa3  =  4.3456588745e+02, /* 0x43d9486f */
-sa4  =  6.4538726807e+02, /* 0x442158c9 */
-sa5  =  4.2900814819e+02, /* 0x43d6810b */
-sa6  =  1.0863500214e+02, /* 0x42d9451f */
-sa7  =  6.5702495575e+00, /* 0x40d23f7c */
-sa8  = -6.0424413532e-02, /* 0xbd777f97 */
+erx  =  8.42697144e-01F, /*  0x1.af7600p-1.  erf(1) rounded to 16 bits. */
+pa0  =  3.64939137e-06F, /*  0x1.e9d022p-19 */
+pa1  =  4.15109694e-01F, /*  0x1.a91284p-2 */
+pa2  = -1.65179938e-01F, /* -0x1.5249dcp-3 */
+pa3  =  1.10914491e-01F, /*  0x1.c64e46p-4 */
+qa1  =  6.02074385e-01F, /*  0x1.344318p-1 */
+qa2  =  5.35934687e-01F, /*  0x1.126608p-1 */
+qa3  =  1.68576106e-01F, /*  0x1.593e6ep-3 */
+qa4  =  5.62181212e-02F, /*  0x1.cc89f2p-5 */
 /*
- * Coefficients for approximation to  erfc in [1/.35,28]
+ * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
  */
-rb0  = -9.8649431020e-03, /* 0xbc21a092 */
-rb1  = -7.9928326607e-01, /* 0xbf4c9dd4 */
-rb2  = -1.7757955551e+01, /* 0xc18e104b */
-rb3  = -1.6063638306e+02, /* 0xc320a2ea */
-rb4  = -6.3756646729e+02, /* 0xc41f6441 */
-rb5  = -1.0250950928e+03, /* 0xc480230b */
-rb6  = -4.8351919556e+02, /* 0xc3f1c275 */
-sb1  =  3.0338060379e+01, /* 0x41f2b459 */
-sb2  =  3.2579251099e+02, /* 0x43a2e571 */
-sb3  =  1.5367296143e+03, /* 0x44c01759 */
-sb4  =  3.1998581543e+03, /* 0x4547fdbb */
-sb5  =  2.5530502930e+03, /* 0x451f90ce */
-sb6  =  4.7452853394e+02, /* 0x43ed43a7 */
-sb7  = -2.2440952301e+01; /* 0xc1b38712 */
+ra0  = -9.87132732e-03F, /* -0x1.4376b2p-7 */
+ra1  = -5.53605914e-01F, /* -0x1.1b723cp-1 */
+ra2  = -2.17589188e+00F, /* -0x1.1683a0p+1 */
+ra3  = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
+sa1  =  5.45995426e+00F, /*  0x1.5d6fe4p+2 */
+sa2  =  6.69798088e+00F, /*  0x1.acabb8p+2 */
+sa3  =  1.43113089e+00F, /*  0x1.6e5e98p+0 */
+sa4  = -5.77397496e-02F, /* -0x1.d90108p-5 */
+/*
+ * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
+ */
+rb0  = -9.86494310e-03F, /* -0x1.434124p-7 */
+rb1  = -6.25171244e-01F, /* -0x1.401672p-1 */
+rb2  = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
+rb3  = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
+rb4  = -9.53764343e+00F, /* -0x1.313460p+3 */
+sb1  =  1.26884899e+01F, /*  0x1.96081cp+3 */
+sb2  =  4.51839523e+01F, /*  0x1.6978bcp+5 */
+sb3  =  4.72810211e+01F, /*  0x1.7a3f88p+5 */
+sb4  =  8.93033314e+00F; /*  0x1.1dc54ap+3 */
 
 float
 erff(float x)
@@ -107,43 +91,37 @@ erff(float x)
 	}
 
 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
-	    if(ix < 0x31800000) { 	/* |x|<2**-28 */
-	        if (ix < 0x04000000)
-		    /*avoid underflow */
-		    return (float)0.125*((float)8.0*x+efx8*x);
+	    if(ix < 0x38800000) { 	/* |x|<2**-14 */
+	        if (ix < 0x04000000)	/* |x|<0x1p-119 */
+		    return (8*x+efx8*x)/8;	/* avoid spurious underflow */
 		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 */
+	if (ix >= 0x40800000) {		/* inf>|x|>=4 */
 	    if(hx>=0) return one-tiny; else return tiny-one;
 	}
 	x = fabsf(x);
  	s = one/(x*x);
 	if(ix< 0x4036DB6E) {	/* |x| < 1/0.35 */
-	    R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
-				ra5+s*(ra6+s*ra7))))));
-	    S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
-				sa5+s*(sa6+s*(sa7+s*sa8)))))));
+	    R=ra0+s*(ra1+s*(ra2+s*ra3));
+	    S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
 	} else {	/* |x| >= 1/0.35 */
-	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
-				rb5+s*rb6)))));
-	    S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
-				sb5+s*(sb6+s*sb7))))));
+	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
+	    S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
 	}
-	GET_FLOAT_WORD(ix,x);
-	SET_FLOAT_WORD(z,ix&0xfffff000);
-	r  =  __ieee754_expf(-z*z-(float)0.5625)*__ieee754_expf((z-x)*(z+x)+R/S);
+	SET_FLOAT_WORD(z,hx&0xffffe000);
+	r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 	if(hx>=0) return one-r/x; else return  r/x-one;
 }
 
@@ -160,11 +138,11 @@ erfcf(float x)
 	}
 
 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
-	    if(ix < 0x23800000)  	/* |x|<2**-56 */
+	    if(ix < 0x33800000)  	/* |x|<2**-24 */
 		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);
@@ -176,33 +154,27 @@ 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 {
 		z = erx+P/Q; return one+z;
 	    }
 	}
-	if (ix < 0x41e00000) {		/* |x|<28 */
+	if (ix < 0x41300000) {		/* |x|<11 */
 	    x = fabsf(x);
  	    s = one/(x*x);
 	    if(ix< 0x4036DB6D) {	/* |x| < 1/.35 ~ 2.857143*/
-	        R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
-				ra5+s*(ra6+s*ra7))))));
-	        S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
-				sa5+s*(sa6+s*(sa7+s*sa8)))))));
+	        R=ra0+s*(ra1+s*(ra2+s*ra3));
+	        S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
 	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
-		if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
-	        R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
-				rb5+s*rb6)))));
-	        S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
-				sb5+s*(sb6+s*sb7))))));
+		if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
+	        R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
+		S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
 	    }
-	    GET_FLOAT_WORD(ix,x);
-	    SET_FLOAT_WORD(z,ix&0xfffff000);
-	    r  =  __ieee754_expf(-z*z-(float)0.5625)*
-			__ieee754_expf((z-x)*(z+x)+R/S);
+	    SET_FLOAT_WORD(z,hx&0xffffe000);
+	    r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 	    if(hx>0) return r/x; else return two-r/x;
 	} else {
 	    if(hx>0) return tiny*tiny; else return two-tiny;