* Use 9 digits instead of 11 digits in efx and efx8.

* Update the domain and range of comments for the polynomial
  approximations, including using the the correct variable names
  (e.g., pp(x) instead of p(x)).

* Use hex values of the form 0x3e0375d4 instead of 0x1.06eba8p-3,
  which was obtained from printf("%.6a").

* In the domain [0.84375, 1.25], qa(x) can be reduced from a 4th
  order polynomial to 3rd order.

* In the domain [1.25,1/0.35], sa(x) can be reduced from a 4th
  order polynomial to 3rd order.

* In the domain [1/0.35, 11], the 4th order polynomials rb(x) and
  sb(x) can be reduced to 2nd and 3rd order, respectively.

1 files changed, 49 insertions, 54 deletions

diff --git a/lib/msun/src/s_erff.c b/lib/msun/src/s_erff.c
index abbf6707fe90..d6cfbd227678 100644
--- a/lib/msun/src/s_erff.c
+++ b/lib/msun/src/s_erff.c
@@ -32,55 +32,50 @@ erx = 8.42697144e-01,			/* 0x3f57bb00 */
  * expansion of erf(x) is used.  The magnitude of the first neglected
  * terms is less than 2**-42.
  */
-efx =  1.2837916613e-01, /* 0x3e0375d4 */
-efx8=  1.0270333290e+00, /* 0x3f8375d4 */
+efx = 1.28379166e-01, /* 0x3e0375d4 */
+efx8= 1.02703333e+00, /* 0x3f8375d4 */
 /*
- *  Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
- *  |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
+ * Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]:
+ * |(erf(x) - x)/x - pp(x)/qq(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 */
+pp0  =  1.28379166e-01, /* 0x3e0375d4 */
+pp1  = -3.36030394e-01, /* 0xbeac0c2d */
+pp2  = -1.86261395e-03, /* 0xbaf422f4 */
+qq1  =  3.12324315e-01, /* 0x3e9fe8f9 */
+qq2  =  2.16070414e-02, /* 0x3cb10140 */
+qq3  = -1.98859372e-03, /* 0xbb025311 */
 /*
- * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
- * |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
+ * Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]:
+ * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31
  */
-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 */
+pa0  =  3.65041046e-06, /* 0x3674f993 */
+pa1  =  4.15109307e-01, /* 0x3ed48935 */
+pa2  = -2.09395722e-01, /* 0xbe566bd5 */
+pa3  =  8.67677554e-02, /* 0x3db1b34b */
+qa1  =  4.95560974e-01, /* 0x3efdba2b */
+qa2  =  3.71248513e-01, /* 0x3ebe1449 */
+qa3  =  3.92478965e-02, /* 0x3d20c267 */
 /*
- * 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
+ * Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28
  */
-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 */
+ra0  = -9.88156721e-03, /* 0xbc21e64c */
+ra1  = -5.43658376e-01, /* 0xbf0b2d32 */
+ra2  = -1.66828310e+00, /* 0xbfd58a4d */
+ra3  = -6.91554189e-01, /* 0xbf3109b2 */
+sa1  =  4.48581553e+00, /* 0x408f8bcd */
+sa2  =  4.10799170e+00, /* 0x408374ab */
+sa3  =  5.53855181e-01, /* 0x3f0dc974 */
 /*
- * 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
+ * Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30
  */
-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 */
+rb0  = -9.86496918e-03, /* 0xbc21a0ae */
+rb1  = -5.48049808e-01, /* 0xbf0c4cfe */
+rb2  = -1.84115684e+00, /* 0xbfebab07 */
+sb1  =  4.87132740e+00, /* 0x409be1ea */
+sb2  =  3.04982710e+00, /* 0x4043305e */
+sb3  = -7.61900663e-01; /* 0xbf430bec */
 
 float
 erff(float x)
@@ -109,7 +104,7 @@ erff(float x)
 	if(ix < 0x3fa00000) {		/* 0.84375 <= |x| < 1.25 */
 	    s = fabsf(x)-one;
 	    P = pa0+s*(pa1+s*(pa2+s*pa3));
-	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
+	    Q = one+s*(qa1+s*(qa2+s*qa3));
 	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 	}
 	if (ix >= 0x40800000) {		/* inf>|x|>=4 */
@@ -117,12 +112,12 @@ erff(float x)
 	}
 	x = fabsf(x);
  	s = one/(x*x);
-	if(ix< 0x4036DB6E) {	/* |x| < 1/0.35 */
+	if(ix< 0x4036db8c) {	/* |x| < 2.85715 ~ 1/0.35 */
 	    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=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
+	    S=one+s*(sa1+s*(sa2+s*sa3));
+	} else {	/* |x| >= 2.85715 ~ 1/0.35 */
+	    R=rb0+s*(rb1+s*rb2);
+	    S=one+s*(sb1+s*(sb2+s*sb3));
 	}
 	SET_FLOAT_WORD(z,hx&0xffffe000);
 	r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
@@ -159,7 +154,7 @@ erfcf(float x)
 	if(ix < 0x3fa00000) {		/* 0.84375 <= |x| < 1.25 */
 	    s = fabsf(x)-one;
 	    P = pa0+s*(pa1+s*(pa2+s*pa3));
-	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
+	    Q = one+s*(qa1+s*(qa2+s*qa3));
 	    if(hx>=0) {
 	        z  = one-erx; return z - P/Q;
 	    } else {
@@ -169,13 +164,13 @@ erfcf(float x)
 	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=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
-	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
+	    if(ix< 0x4036db8c) {	/* |x| < 2.85715 ~ 1/.35 */
+		R=ra0+s*(ra1+s*(ra2+s*ra3));
+		S=one+s*(sa1+s*(sa2+s*sa3));
+	    } else {			/* |x| >= 2.85715 ~ 1/.35 */
 		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)));
+		R=rb0+s*(rb1+s*rb2);
+		S=one+s*(sb1+s*(sb2+s*sb3));
 	    }
 	    SET_FLOAT_WORD(z,hx&0xffffe000);
 	    r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);