libm: Sync with FreeBSD (gains 6 long double functions)
[dragonfly.git] / lib / libm / src / s_erff.c
index eece494..d71f8c4 100644 (file)
  * is preserved.
  * ====================================================
  *
- * $FreeBSD: head/lib/msun/src/s_erff.c 254969 2013-08-27 19:46:56Z kargl $
+ * $FreeBSD: head/lib/msun/src/s_erff.c 268590 2014-07-13 16:24:16Z kargl $
  */
 
 
 #include "math.h"
 #include "math_private.h"
 
+/* XXX Prevent compilers from erroneously constant folding: */
+static const volatile float tiny = 1e-30;
+
 static const float
-tiny       = 1e-30,
-half=  5.0000000000e-01, /* 0x3F000000 */
-one =  1.0000000000e+00, /* 0x3F800000 */
-two =  2.0000000000e+00, /* 0x40000000 */
+half= 0.5,
+one = 1,
+two = 2,
+erx = 8.42697144e-01,                  /* 0x3f57bb00 */
 /*
- * Coefficients for approximation to erf on [0,0.84375]
+ * In the domain [0, 2**-14], only the first term in the power series
+ * 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
  */
-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 */
+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)
@@ -85,9 +84,9 @@ erff(float x)
        float R,S,P,Q,s,y,z,r;
        GET_FLOAT_WORD(hx,x);
        ix = hx&0x7fffffff;
-       if(ix>=0x7f800000) {            /* erf(nan)=nan */
+       if(ix>=0x7f800000) {            /* erff(nan)=nan */
            i = ((u_int32_t)hx>>31)<<1;
-           return (float)(1-i)+one/x;  /* erf(+-inf)=+-1 */
+           return (float)(1-i)+one/x;  /* erff(+-inf)=+-1 */
        }
 
        if(ix < 0x3f580000) {           /* |x|<0.84375 */
@@ -105,7 +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 */
@@ -113,12 +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);
@@ -132,8 +131,8 @@ erfcf(float x)
        float R,S,P,Q,s,y,z,r;
        GET_FLOAT_WORD(hx,x);
        ix = hx&0x7fffffff;
-       if(ix>=0x7f800000) {                    /* erfc(nan)=nan */
-                                               /* erfc(+-inf)=0,2 */
+       if(ix>=0x7f800000) {                    /* erfcf(nan)=nan */
+                                               /* erfcf(+-inf)=0,2 */
            return (float)(((u_int32_t)hx>>31)<<1)+one/x;
        }
 
@@ -155,7 +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 {
@@ -165,13 +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);