Update libm to the upstream head (2014.03.13)

The most relevant changes are the ones in s_erff.c which should improve
the overall speed of exececution and the maximum error in ULP.

Change-Id: I7bd867f902db418db67f02cd58578624357b1ee8

4 files changed, 72 insertions, 100 deletions

diff --git a/libm/upstream-freebsd/lib/msun/src/math_private.h b/libm/upstream-freebsd/lib/msun/src/math_private.h
index 8ebc7fb..8af2c65 100644
--- a/libm/upstream-freebsd/lib/msun/src/math_private.h
+++ b/libm/upstream-freebsd/lib/msun/src/math_private.h
@@ -39,7 +39,7 @@
  */
 
 #ifdef __arm__
-#if defined(__VFP_FP__)
+#if defined(__VFP_FP__) || defined(__ARM_EABI__)
 #define	IEEE_WORD_ORDER	BYTE_ORDER
 #else
 #define	IEEE_WORD_ORDER	BIG_ENDIAN
diff --git a/libm/upstream-freebsd/lib/msun/src/s_erff.c b/libm/upstream-freebsd/lib/msun/src/s_erff.c
index a44e135..a3579f1 100644
--- a/libm/upstream-freebsd/lib/msun/src/s_erff.c
+++ b/libm/upstream-freebsd/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]
  */
 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;
diff --git a/libm/upstream-freebsd/lib/msun/src/s_fma.c b/libm/upstream-freebsd/lib/msun/src/s_fma.c
index 452bece..b1066c2 100644
--- a/libm/upstream-freebsd/lib/msun/src/s_fma.c
+++ b/libm/upstream-freebsd/lib/msun/src/s_fma.c
@@ -117,7 +117,7 @@ add_and_denormalize(double a, double b, int scale)
 	if (sum.lo != 0) {
 		EXTRACT_WORD64(hibits, sum.hi);
 		bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
-		if (bits_lost != 1 ^ (int)(hibits & 1)) {
+		if ((bits_lost != 1) ^ (int)(hibits & 1)) {
 			/* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
 			EXTRACT_WORD64(lobits, sum.lo);
 			hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
diff --git a/libm/upstream-freebsd/lib/msun/src/s_fmal.c b/libm/upstream-freebsd/lib/msun/src/s_fmal.c
index 9271901..d435f4f 100644
--- a/libm/upstream-freebsd/lib/msun/src/s_fmal.c
+++ b/libm/upstream-freebsd/lib/msun/src/s_fmal.c
@@ -113,7 +113,7 @@ add_and_denormalize(long double a, long double b, int scale)
 	if (sum.lo != 0) {
 		u.e = sum.hi;
 		bits_lost = -u.bits.exp - scale + 1;
-		if (bits_lost != 1 ^ (int)(u.bits.manl & 1))
+		if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
 			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
 	}
 	return (ldexp(sum.hi, scale));