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
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 @@
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 @@
}
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 @@
}
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 @@
}
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 @@
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 @@
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));
