math: erf and erfc cleanup

common part of erf and erfc was put in a separate function which
saved some space and the new code is using unsigned arithmetics

erfcf had a bug: for some inputs in [7.95,8] the result had
more than 60ulp error: in expf(-z*z - 0.5625f) the argument
must be exact but not enough lowbits of z were zeroed,
-SET_FLOAT_WORD(z, ix&0xfffff000);
+SET_FLOAT_WORD(z, ix&0xffffe000);
fixed the issue

src/math/erf.c

@@ -106,13 +106,10 @@
 #include "libm.h"
 
 static const double
-tiny = 1e-300,
-/* c = (float)0.84506291151 */
 erx  = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 /*
  * Coefficients for approximation to  erf on [0,0.84375]
  */
-efx  =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 efx8 =  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
@@ -177,52 +174,31 @@ sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
 
-double erf(double x)
+static double erfc1(double x)
 {
-	int32_t hx,ix,i;
-	double R,S,P,Q,s,y,z,r;
+	double s,P,Q;
+
+	s = fabs(x) - 1;
+	P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
+	Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+	return 1 - erx - P/Q;
+}
+
+static double erfc2(uint32_t ix, double x)
+{
+	double s,z,R,S;
+
+	if (ix < 0x3ff40000)  /* |x| < 1.25 */
+		return erfc1(x);
 
-	GET_HIGH_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	if (ix >= 0x7ff00000) {
-		/* erf(nan)=nan, erf(+-inf)=+-1 */
-		i = ((uint32_t)hx>>31)<<1;
-		return (double)(1-i) + 1.0/x;
-	}
-	if (ix < 0x3feb0000) {  /* |x|<0.84375 */
-		if (ix < 0x3e300000) {  /* |x|<2**-28 */
-			if (ix < 0x00800000)
-				/* avoid underflow */
-				return 0.125*(8.0*x + efx8*x);
-			return x + efx*x;
-		}
-		z = x*x;
-		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
-		s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
-		y = r/s;
-		return x + x*y;
-	}
-	if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabs(x)-1.0;
-		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
-		Q = 1.0+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
-		if (hx >= 0)
-			return erx + P/Q;
-		return -erx - P/Q;
-	}
-	if (ix >= 0x40180000) {  /* inf > |x| >= 6 */
-		if (hx >= 0)
-			return 1.0 - tiny;
-		return tiny - 1.0;
-	}
 	x = fabs(x);
-	s = 1.0/(x*x);
-	if (ix < 0x4006DB6E) {  /* |x| < 1/0.35 */
+	s = 1/(x*x);
+	if (ix < 0x4006db6d) {  /* |x| < 1/.35 ~ 2.85714 */
 		R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 		     ra5+s*(ra6+s*ra7))))));
 		S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 		     sa5+s*(sa6+s*(sa7+s*sa8)))))));
-	} else {                /* |x| >= 1/0.35 */
+	} else {                /* |x| > 1/.35 */
 		R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 		     rb5+s*rb6)))));
 		S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
@@ -230,22 +206,52 @@ double erf(double x)
 	}
 	z = x;
 	SET_LOW_WORD(z,0);
-	r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
-	if (hx >= 0)
-		return 1.0 - r/x;
-	return r/x - 1.0;
+	return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x;
+}
+
+double erf(double x)
+{
+	double r,s,z,y;
+	uint32_t ix;
+	int sign;
+
+	GET_HIGH_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
+	if (ix >= 0x7ff00000) {
+		/* erf(nan)=nan, erf(+-inf)=+-1 */
+		return 1-2*sign + 1/x;
+	}
+	if (ix < 0x3feb0000) {  /* |x| < 0.84375 */
+		if (ix < 0x3e300000) {  /* |x| < 2**-28 */
+			/* avoid underflow */
+			return 0.125*(8*x + efx8*x);
+		}
+		z = x*x;
+		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
+		s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+		y = r/s;
+		return x + x*y;
+	}
+	if (ix < 0x40180000)  /* 0.84375 <= |x| < 6 */
+		y = 1 - erfc2(ix,x);
+	else
+		y = 1 - 0x1p-1022;
+	return sign ? -y : y;
 }
 
 double erfc(double x)
 {
-	int32_t hx,ix;
-	double R,S,P,Q,s,y,z,r;
+	double r,s,z,y;
+	uint32_t ix;
+	int sign;
 
-	GET_HIGH_WORD(hx, x);
-	ix = hx & 0x7fffffff;
+	GET_HIGH_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
 	if (ix >= 0x7ff00000) {
 		/* erfc(nan)=nan, erfc(+-inf)=0,2 */
-		return (double)(((uint32_t)hx>>31)<<1) + 1.0/x;
+		return 2*sign + 1/x;
 	}
 	if (ix < 0x3feb0000) {  /* |x| < 0.84375 */
 		if (ix < 0x3c700000)  /* |x| < 2**-56 */
@@ -254,50 +260,13 @@ double erfc(double x)
 		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 		s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 		y = r/s;
-		if (hx < 0x3fd00000) {  /* x < 1/4 */
+		if (sign || ix < 0x3fd00000) {  /* x < 1/4 */
 			return 1.0 - (x+x*y);
-		} else {
-			r = x*y;
-			r += x - 0.5;
-			return 0.5 - r ;
-		}
-	}
-	if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabs(x)-1.0;
-		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
-		Q = 1.0+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
-		if (hx >= 0) {
-			z = 1.0-erx;
-			return z - P/Q;
-		} else {
-			z = erx+P/Q;
-			return 1.0 + z;
 		}
+		return 0.5 - (x - 0.5 + x*y);
 	}
-	if (ix < 0x403c0000) {  /* |x| < 28 */
-		x = fabs(x);
-		s = 1.0/(x*x);
-		if (ix < 0x4006DB6D) {  /* |x| < 1/.35 ~ 2.857143*/
-			R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
-			     ra5+s*(ra6+s*ra7))))));
-			S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
-			     sa5+s*(sa6+s*(sa7+s*sa8)))))));
-		} else {                /* |x| >= 1/.35 ~ 2.857143 */
-			if (hx < 0 && ix >= 0x40180000)  /* x < -6 */
-				return 2.0 - tiny;
-			R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
-			     rb5+s*rb6)))));
-			S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
-			     sb5+s*(sb6+s*sb7))))));
-		}
-		z = x;
-		SET_LOW_WORD(z, 0);
-		r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
-		if (hx > 0)
-			return r/x;
-		return 2.0 - r/x;
+	if (ix < 0x403c0000) {  /* 0.84375 <= |x| < 28 */
+		return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
 	}
-	if (hx > 0)
-		return tiny*tiny;
-	return 2.0 - tiny;
+	return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022;
 }

src/math/erff.c

@@ -16,13 +16,10 @@
 #include "libm.h"
 
 static const float
-tiny = 1e-30,
-/* c = (subfloat)0.84506291151 */
 erx  =  8.4506291151e-01, /* 0x3f58560b */
 /*
  * 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 */
@@ -87,47 +84,26 @@ sb5  =  2.5530502930e+03, /* 0x451f90ce */
 sb6  =  4.7452853394e+02, /* 0x43ed43a7 */
 sb7  = -2.2440952301e+01; /* 0xc1b38712 */
 
-float erff(float x)
+static float erfc1(float x)
 {
-	int32_t hx,ix,i;
-	float R,S,P,Q,s,y,z,r;
+	float s,P,Q;
+
+	s = fabsf(x) - 1;
+	P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
+	Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+	return 1 - erx - P/Q;
+}
+
+static float erfc2(uint32_t ix, float x)
+{
+	float s,z,R,S;
+
+	if (ix < 0x3fa00000)  /* |x| < 1.25 */
+		return erfc1(x);
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	if (ix >= 0x7f800000) {
-		/* erf(nan)=nan, erf(+-inf)=+-1 */
-		i = ((uint32_t)hx>>31)<<1;
-		return (float)(1-i)+1.0f/x;
-	}
-	if (ix < 0x3f580000) {  /* |x| < 0.84375 */
-		if (ix < 0x31800000) {  /* |x| < 2**-28 */
-			if (ix < 0x04000000)
-				/*avoid underflow */
-				return 0.125f*(8.0f*x + efx8*x);
-			return x + efx*x;
-		}
-		z = x*x;
-		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
-		s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
-		y = r/s;
-		return x + x*y;
-	}
-	if (ix < 0x3fa00000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabsf(x)-1.0f;
-		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
-		Q = 1.0f+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
-		if (hx >= 0)
-			return erx + P/Q;
-		return -erx - P/Q;
-	}
-	if (ix >= 0x40c00000) {  /* inf > |x| >= 6 */
-		if (hx >= 0)
-			return 1.0f - tiny;
-		return tiny - 1.0f;
-	}
 	x = fabsf(x);
-	s = 1.0f/(x*x);
-	if (ix < 0x4036DB6E) {   /* |x| < 1/0.35 */
+	s = 1/(x*x);
+	if (ix < 0x4036db6d) {   /* |x| < 1/0.35 */
 		R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 		     ra5+s*(ra6+s*ra7))))));
 		S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
@@ -139,23 +115,53 @@ float erff(float x)
 		     sb5+s*(sb6+s*sb7))))));
 	}
 	GET_FLOAT_WORD(ix, x);
-	SET_FLOAT_WORD(z, ix&0xfffff000);
-	r = expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S);
-	if (hx >= 0)
-		return 1.0f - r/x;
-	return  r/x - 1.0f;
+	SET_FLOAT_WORD(z, ix&0xffffe000);
+	return expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S)/x;
+}
+
+float erff(float x)
+{
+	float r,s,z,y;
+	uint32_t ix;
+	int sign;
+
+	GET_FLOAT_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
+	if (ix >= 0x7f800000) {
+		/* erf(nan)=nan, erf(+-inf)=+-1 */
+		return 1-2*sign + 1/x;
+	}
+	if (ix < 0x3f580000) {  /* |x| < 0.84375 */
+		if (ix < 0x31800000) {  /* |x| < 2**-28 */
+			/*avoid underflow */
+			return 0.125f*(8*x + efx8*x);
+		}
+		z = x*x;
+		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
+		s = 1+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+		y = r/s;
+		return x + x*y;
+	}
+	if (ix < 0x40c00000)  /* |x| < 6 */
+		y = 1 - erfc2(ix,x);
+	else
+		y = 1 - 0x1p-120f;
+	return sign ? -y : y;
 }
 
 float erfcf(float x)
 {
-	int32_t hx,ix;
-	float R,S,P,Q,s,y,z,r;
+	float r,s,z,y;
+	uint32_t ix;
+	int sign;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
+	GET_FLOAT_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
 	if (ix >= 0x7f800000) {
 		/* erfc(nan)=nan, erfc(+-inf)=0,2 */
-		return (float)(((uint32_t)hx>>31)<<1) + 1.0f/x;
+		return 2*sign + 1/x;
 	}
 
 	if (ix < 0x3f580000) {  /* |x| < 0.84375 */
@@ -165,50 +171,12 @@ float erfcf(float x)
 		r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 		s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 		y = r/s;
-		if (hx < 0x3e800000) {  /* x<1/4 */
+		if (sign || ix < 0x3e800000)  /* x < 1/4 */
 			return 1.0f - (x+x*y);
-		} else {
-			r = x*y;
-			r += (x-0.5f);
-			return 0.5f - r ;
-		}
-	}
-	if (ix < 0x3fa00000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabsf(x)-1.0f;
-		P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
-		Q = 1.0f+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
-		if(hx >= 0) {
-			z = 1.0f - erx;
-			return z - P/Q;
-		} else {
-			z = erx + P/Q;
-			return 1.0f + z;
-		}
+		return 0.5f - (x - 0.5f + x*y);
 	}
 	if (ix < 0x41e00000) {  /* |x| < 28 */
-		x = fabsf(x);
-		s = 1.0f/(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 = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
-			     sa5+s*(sa6+s*(sa7+s*sa8)))))));
-		} else {                /* |x| >= 1/.35 ~ 2.857143 */
-			if (hx < 0 && ix >= 0x40c00000) /* x < -6 */
-				return 2.0f-tiny;
-			R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
-			     rb5+s*rb6)))));
-			S = 1.0f+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
-			     sb5+s*(sb6+s*sb7))))));
-		}
-		GET_FLOAT_WORD(ix, x);
-		SET_FLOAT_WORD(z, ix&0xfffff000);
-		r = expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S);
-		if (hx > 0)
-			return r/x;
-		return 2.0f - r/x;
+		return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
 	}
-	if (hx > 0)
-		return tiny*tiny;
-	return 2.0f - tiny;
+	return sign ? 2 - 0x1p-120f : 0x1p-120f*0x1p-120f;
 }

src/math/erfl.c

@@ -107,15 +107,11 @@ long double erfl(long double x)
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
 static const long double
-tiny = 1e-4931L,
-/* c = (float)0.84506291151 */
 erx = 0.845062911510467529296875L,
 
 /*
  * Coefficients for approximation to  erf on [0,0.84375]
  */
-/* 2/sqrt(pi) - 1 */
-efx = 1.2837916709551257389615890312154517168810E-1L,
 /* 8 * (2/sqrt(pi) - 1) */
 efx8 = 1.0270333367641005911692712249723613735048E0L,
 pp[6] = {
@@ -239,84 +235,108 @@ sc[] = {
 	/* 1.000000000000000000000000000000000000000E0 */
 };
 
-long double erfl(long double x)
+static long double erfc1(long double x)
 {
-	long double R, S, P, Q, s, y, z, r;
-	int32_t ix, i;
-	uint32_t se, i0, i1;
+	long double s,P,Q;
 
-	GET_LDOUBLE_WORDS(se, i0, i1, x);
-	ix = se & 0x7fff;
+	s = fabsl(x) - 1;
+	P = pa[0] + s * (pa[1] + s * (pa[2] +
+	     s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
+	Q = qa[0] + s * (qa[1] + s * (qa[2] +
+	     s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
+	return 1 - erx - P / Q;
+}
 
-	if (ix >= 0x7fff) {  /* erf(nan)=nan */
-		i = ((se & 0xffff) >> 15) << 1;
-		return (long double)(1 - i) + 1.0 / x;  /* erf(+-inf)=+-1 */
-	}
+static long double erfc2(uint32_t ix, long double x)
+{
+	long double s,z,R,S;
+	uint32_t i0,i1;
 
-	ix = (ix << 16) | (i0 >> 16);
-	if (ix < 0x3ffed800) {  /* |x| < 0.84375 */
-		if (ix < 0x3fde8000) {  /* |x| < 2**-33 */
-			if (ix < 0x00080000)
-				return 0.125 * (8.0 * x + efx8 * x);  /* avoid underflow */
-			return x + efx * x;
-		}
-		z = x * x;
-		r = pp[0] + z * (pp[1] +
-		     z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
-		s = qq[0] + z * (qq[1] +
-		     z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
-		y = r / s;
-		return x + x * y;
-	}
-	if (ix < 0x3fffa000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabsl(x) - 1.0;
-		P = pa[0] + s * (pa[1] + s * (pa[2] +
-		     s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
-		Q = qa[0] + s * (qa[1] + s * (qa[2] +
-		     s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
-		if ((se & 0x8000) == 0)
-			return erx + P / Q;
-		return -erx - P / Q;
-	}
-	if (ix >= 0x4001d555) {  /* inf > |x| >= 6.6666259765625 */
-		if ((se & 0x8000) == 0)
-			return 1.0 - tiny;
-		return tiny - 1.0;
+	if (ix < 0x3fffa000)  /* 0.84375 <= |x| < 1.25 */
+		return erfc1(x);
+
+	x = fabsl(x);
+	s = 1 / (x * x);
+	if (ix < 0x4000b6db) {  /* 1.25 <= |x| < 2.857 ~ 1/.35 */
+		R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
+		     s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
+		S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
+		     s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
+	} else {  /* 2.857 <= |x| */
+		R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
+		     s * (rb[5] + s * (rb[6] + s * rb[7]))))));
+		S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
+		     s * (sb[5] + s * (sb[6] + s))))));
 	}
-	x = fabsl (x);
-	s = 1.0 / (x * x);
 	if (ix < 0x4000b6db) {  /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
 		R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
 		     s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
 		S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
 		     s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
-	} else { /* 2.857 <= |x| < 6.667 */
+	} else if (ix < 0x4001d555) {  /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */
 		R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
 		     s * (rb[5] + s * (rb[6] + s * rb[7]))))));
 		S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
 		     s * (sb[5] + s * (sb[6] + s))))));
+	} else { /* 107 > |x| >= 6.666 */
+		R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
+		     s * (rc[4] + s * rc[5]))));
+		S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
+		     s * (sc[4] + s))));
 	}
 	z = x;
-	GET_LDOUBLE_WORDS(i, i0, i1, z);
+	GET_LDOUBLE_WORDS(ix, i0, i1, z);
 	i1 = 0;
-	SET_LDOUBLE_WORDS(z, i, i0, i1);
-	r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S);
-	if ((se & 0x8000) == 0)
-		return 1.0 - r / x;
-	return r / x - 1.0;
+	i0 &= 0xffffff00;
+	SET_LDOUBLE_WORDS(z, ix, i0, i1);
+	return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x;
 }
 
-long double erfcl(long double x)
+long double erfl(long double x)
 {
-	int32_t hx, ix;
-	long double R, S, P, Q, s, y, z, r;
-	uint32_t se, i0, i1;
+	long double r, s, z, y;
+	uint32_t i0, i1, ix;
+	int sign;
 
-	GET_LDOUBLE_WORDS(se, i0, i1, x);
-	ix = se & 0x7fff;
-	if (ix >= 0x7fff) {  /* erfc(nan) = nan, erfc(+-inf) = 0,2 */
-		return (long double)(((se & 0xffff) >> 15) << 1) + 1.0 / x;
+	GET_LDOUBLE_WORDS(ix, i0, i1, x);
+	sign = ix >> 15;
+	ix &= 0x7fff;
+	if (ix == 0x7fff) {
+		/* erf(nan)=nan, erf(+-inf)=+-1 */
+		return 1 - 2*sign + 1/x;
 	}
+	ix = (ix << 16) | (i0 >> 16);
+	if (ix < 0x3ffed800) {  /* |x| < 0.84375 */
+		if (ix < 0x3fde8000) {  /* |x| < 2**-33 */
+			return 0.125 * (8 * x + efx8 * x);  /* avoid underflow */
+		}
+		z = x * x;
+		r = pp[0] + z * (pp[1] +
+		     z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
+		s = qq[0] + z * (qq[1] +
+		     z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
+		y = r / s;
+		return x + x * y;
+	}
+	if (ix < 0x4001d555)  /* |x| < 6.6666259765625 */
+		y = 1 - erfc2(ix,x);
+	else
+		y = 1 - 0x1p-16382L;
+	return sign ? -y : y;
+}
+
+long double erfcl(long double x)
+{
+	long double r, s, z, y;
+	uint32_t i0, i1, ix;
+	int sign;
+
+	GET_LDOUBLE_WORDS(ix, i0, i1, x);
+	sign = ix>>15;
+	ix &= 0x7fff;
+	if (ix == 0x7fff)
+		/* erfc(nan) = nan, erfc(+-inf) = 0,2 */
+		return 2*sign + 1/x;
 
 	ix = (ix << 16) | (i0 >> 16);
 	if (ix < 0x3ffed800) {  /* |x| < 0.84375 */
@@ -330,57 +350,10 @@ long double erfcl(long double x)
 		y = r / s;
 		if (ix < 0x3ffd8000) /* x < 1/4 */
 			return 1.0 - (x + x * y);
-		r = x * y;
-		r += x - 0.5L;
-		return 0.5L - r;
-	}
-	if (ix < 0x3fffa000) {  /* 0.84375 <= |x| < 1.25 */
-		s = fabsl(x) - 1.0;
-		P = pa[0] + s * (pa[1] + s * (pa[2] +
-		     s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
-		Q = qa[0] + s * (qa[1] + s * (qa[2] +
-		     s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
-		if ((se & 0x8000) == 0) {
-			z = 1.0 - erx;
-			return z - P / Q;
-		}
-		z = erx + P / Q;
-		return 1.0 + z;
+		return 0.5 - (x - 0.5 + x * y);
 	}
-	if (ix < 0x4005d600) {  /* |x| < 107 */
-		x = fabsl(x);
-		s = 1.0 / (x * x);
-		if (ix < 0x4000b6db) {  /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */
-			R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
-			     s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
-			S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
-			     s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
-		} else if (ix < 0x4001d555) {  /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */
-			R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
-			     s * (rb[5] + s * (rb[6] + s * rb[7]))))));
-			S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
-			     s * (sb[5] + s * (sb[6] + s))))));
-		} else { /* 107 > |x| >= 6.666 */
-			if (se & 0x8000)
-				return 2.0 - tiny;/* x < -6.666 */
-			R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
-			     s * (rc[4] + s * rc[5]))));
-			S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
-			     s * (sc[4] + s))));
-		}
-		z = x;
-		GET_LDOUBLE_WORDS(hx, i0, i1, z);
-		i1 = 0;
-		i0 &= 0xffffff00;
-		SET_LDOUBLE_WORDS(z, hx, i0, i1);
-		r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S);
-		if ((se & 0x8000) == 0)
-			return r / x;
-		return 2.0 - r / x;
-	}
-
-	if ((se & 0x8000) == 0)
-		return tiny * tiny;
-	return 2.0 - tiny;
+	if (ix < 0x4005d600)  /* |x| < 107 */
+		return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
+	return sign ? 2 - 0x1p-16382L : 0x1p-16382L*0x1p-16382L;
 }
 #endif