lib/libm: Add implementations of erf, erfc, lgamma, tgamma.

9ab94c46 · Damien George · 35270855 · 9ab94c46 · 9ab94c46 · 9ab94c46
Commit 9ab94c46 authored Feb 22, 2015 by Damien George
--- a/lib/libm/erf_lgamma.c
+++ b/lib/libm/erf_lgamma.c
+/*
+ * This file is part of the Micro Python project, http://micropython.org/
+ *
+ * These math functions are taken from newlib-nano-2, the newlib/libm/math
+ * directory, available from https://github.com/32bitmicro/newlib-nano-2.
+ *
+ * Appropriate copyright headers are reproduced below.
+ */
+
+/* erf_lgamma.c -- float version of er_lgamma.c.
+ * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
+ */
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ *
+ */
+
+#include "fdlibm.h"
+
+#define __ieee754_logf logf
+
+#ifdef __STDC__
+static const float 
+#else
+static float 
+#endif
+two23=  8.3886080000e+06, /* 0x4b000000 */
+half=  5.0000000000e-01, /* 0x3f000000 */
+one =  1.0000000000e+00, /* 0x3f800000 */
+pi  =  3.1415927410e+00, /* 0x40490fdb */
+a0  =  7.7215664089e-02, /* 0x3d9e233f */
+a1  =  3.2246702909e-01, /* 0x3ea51a66 */
+a2  =  6.7352302372e-02, /* 0x3d89f001 */
+a3  =  2.0580807701e-02, /* 0x3ca89915 */
+a4  =  7.3855509982e-03, /* 0x3bf2027e */
+a5  =  2.8905137442e-03, /* 0x3b3d6ec6 */
+a6  =  1.1927076848e-03, /* 0x3a9c54a1 */
+a7  =  5.1006977446e-04, /* 0x3a05b634 */
+a8  =  2.2086278477e-04, /* 0x39679767 */
+a9  =  1.0801156895e-04, /* 0x38e28445 */
+a10 =  2.5214456400e-05, /* 0x37d383a2 */
+a11 =  4.4864096708e-05, /* 0x383c2c75 */
+tc  =  1.4616321325e+00, /* 0x3fbb16c3 */
+tf  = -1.2148628384e-01, /* 0xbdf8cdcd */
+/* tt = -(tail of tf) */
+tt  =  6.6971006518e-09, /* 0x31e61c52 */
+t0  =  4.8383611441e-01, /* 0x3ef7b95e */
+t1  = -1.4758771658e-01, /* 0xbe17213c */
+t2  =  6.4624942839e-02, /* 0x3d845a15 */
+t3  = -3.2788541168e-02, /* 0xbd064d47 */
+t4  =  1.7970675603e-02, /* 0x3c93373d */
+t5  = -1.0314224288e-02, /* 0xbc28fcfe */
+t6  =  6.1005386524e-03, /* 0x3bc7e707 */
+t7  = -3.6845202558e-03, /* 0xbb7177fe */
+t8  =  2.2596477065e-03, /* 0x3b141699 */
+t9  = -1.4034647029e-03, /* 0xbab7f476 */
+t10 =  8.8108185446e-04, /* 0x3a66f867 */
+t11 = -5.3859531181e-04, /* 0xba0d3085 */
+t12 =  3.1563205994e-04, /* 0x39a57b6b */
+t13 = -3.1275415677e-04, /* 0xb9a3f927 */
+t14 =  3.3552918467e-04, /* 0x39afe9f7 */
+u0  = -7.7215664089e-02, /* 0xbd9e233f */
+u1  =  6.3282704353e-01, /* 0x3f2200f4 */
+u2  =  1.4549225569e+00, /* 0x3fba3ae7 */
+u3  =  9.7771751881e-01, /* 0x3f7a4bb2 */
+u4  =  2.2896373272e-01, /* 0x3e6a7578 */
+u5  =  1.3381091878e-02, /* 0x3c5b3c5e */
+v1  =  2.4559779167e+00, /* 0x401d2ebe */
+v2  =  2.1284897327e+00, /* 0x4008392d */
+v3  =  7.6928514242e-01, /* 0x3f44efdf */
+v4  =  1.0422264785e-01, /* 0x3dd572af */
+v5  =  3.2170924824e-03, /* 0x3b52d5db */
+s0  = -7.7215664089e-02, /* 0xbd9e233f */
+s1  =  2.1498242021e-01, /* 0x3e5c245a */
+s2  =  3.2577878237e-01, /* 0x3ea6cc7a */
+s3  =  1.4635047317e-01, /* 0x3e15dce6 */
+s4  =  2.6642270386e-02, /* 0x3cda40e4 */
+s5  =  1.8402845599e-03, /* 0x3af135b4 */
+s6  =  3.1947532989e-05, /* 0x3805ff67 */
+r1  =  1.3920053244e+00, /* 0x3fb22d3b */
+r2  =  7.2193557024e-01, /* 0x3f38d0c5 */
+r3  =  1.7193385959e-01, /* 0x3e300f6e */
+r4  =  1.8645919859e-02, /* 0x3c98bf54 */
+r5  =  7.7794247773e-04, /* 0x3a4beed6 */
+r6  =  7.3266842264e-06, /* 0x36f5d7bd */
+w0  =  4.1893854737e-01, /* 0x3ed67f1d */
+w1  =  8.3333335817e-02, /* 0x3daaaaab */
+w2  = -2.7777778450e-03, /* 0xbb360b61 */
+w3  =  7.9365057172e-04, /* 0x3a500cfd */
+w4  = -5.9518753551e-04, /* 0xba1c065c */
+w5  =  8.3633989561e-04, /* 0x3a5b3dd2 */
+w6  = -1.6309292987e-03; /* 0xbad5c4e8 */
+
+#ifdef __STDC__
+static const float zero=  0.0000000000e+00;
+#else
+static float zero=  0.0000000000e+00;
+#endif
+
+#ifdef __STDC__
+	static float sin_pif(float x)
+#else
+	static float sin_pif(x)
+	float x;
+#endif
+{
+	float y,z;
+	__int32_t n,ix;
+
+	GET_FLOAT_WORD(ix,x);
+	ix &= 0x7fffffff;
+
+	if(ix<0x3e800000) return __kernel_sinf(pi*x,zero,0);
+	y = -x;		/* x is assume negative */
+
+    /*
+     * argument reduction, make sure inexact flag not raised if input
+     * is an integer
+     */
+	z = floorf(y);
+	if(z!=y) {				/* inexact anyway */
+	    y  *= (float)0.5;
+	    y   = (float)2.0*(y - floorf(y));	/* y = |x| mod 2.0 */
+	    n   = (__int32_t) (y*(float)4.0);
+	} else {
+            if(ix>=0x4b800000) {
+                y = zero; n = 0;                 /* y must be even */
+            } else {
+                if(ix<0x4b000000) z = y+two23;	/* exact */
+		GET_FLOAT_WORD(n,z);
+		n &= 1;
+                y  = n;
+                n<<= 2;
+            }
+        }
+	switch (n) {
+	    case 0:   y =  __kernel_sinf(pi*y,zero,0); break;
+	    case 1:   
+	    case 2:   y =  __kernel_cosf(pi*((float)0.5-y),zero); break;
+	    case 3:  
+	    case 4:   y =  __kernel_sinf(pi*(one-y),zero,0); break;
+	    case 5:
+	    case 6:   y = -__kernel_cosf(pi*(y-(float)1.5),zero); break;
+	    default:  y =  __kernel_sinf(pi*(y-(float)2.0),zero,0); break;
+	    }
+	return -y;
+}
+
+
+#ifdef __STDC__
+	float __ieee754_lgammaf_r(float x, int *signgamp)
+#else
+	float __ieee754_lgammaf_r(x,signgamp)
+	float x; int *signgamp;
+#endif
+{
+	float t,y,z,nadj = 0.0,p,p1,p2,p3,q,r,w;
+	__int32_t i,hx,ix;
+
+	GET_FLOAT_WORD(hx,x);
+
+    /* purge off +-inf, NaN, +-0, and negative arguments */
+	*signgamp = 1;
+	ix = hx&0x7fffffff;
+	if(ix>=0x7f800000) return x*x;
+	if(ix==0) return one/zero;
+	if(ix<0x1c800000) {	/* |x|<2**-70, return -log(|x|) */
+	    if(hx<0) {
+	        *signgamp = -1;
+	        return -__ieee754_logf(-x);
+	    } else return -__ieee754_logf(x);
+	}
+	if(hx<0) {
+	    if(ix>=0x4b000000) 	/* |x|>=2**23, must be -integer */
+		return one/zero;
+	    t = sin_pif(x);
+	    if(t==zero) return one/zero; /* -integer */
+	    nadj = __ieee754_logf(pi/fabsf(t*x));
+	    if(t<zero) *signgamp = -1;
+	    x = -x;
+	}
+
+    /* purge off 1 and 2 */
+	if (ix==0x3f800000||ix==0x40000000) r = 0;
+    /* for x < 2.0 */
+	else if(ix<0x40000000) {
+	    if(ix<=0x3f666666) { 	/* lgamma(x) = lgamma(x+1)-log(x) */
+		r = -__ieee754_logf(x);
+		if(ix>=0x3f3b4a20) {y = one-x; i= 0;}
+		else if(ix>=0x3e6d3308) {y= x-(tc-one); i=1;}
+	  	else {y = x; i=2;}
+	    } else {
+	  	r = zero;
+	        if(ix>=0x3fdda618) {y=(float)2.0-x;i=0;} /* [1.7316,2] */
+	        else if(ix>=0x3F9da620) {y=x-tc;i=1;} /* [1.23,1.73] */
+		else {y=x-one;i=2;}
+	    }
+	    switch(i) {
+	      case 0:
+		z = y*y;
+		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
+		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
+		p  = y*p1+p2;
+		r  += (p-(float)0.5*y); break;
+	      case 1:
+		z = y*y;
+		w = z*y;
+		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
+		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
+		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
+		p  = z*p1-(tt-w*(p2+y*p3));
+		r += (tf + p); break;
+	      case 2:	
+		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
+		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
+		r += (-(float)0.5*y + p1/p2);
+	    }
+	}
+	else if(ix<0x41000000) { 			/* x < 8.0 */
+	    i = (__int32_t)x;
+	    t = zero;
+	    y = x-(float)i;
+	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
+	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
+	    r = half*y+p/q;
+	    z = one;	/* lgamma(1+s) = log(s) + lgamma(s) */
+	    switch(i) {
+	    case 7: z *= (y+(float)6.0);	/* FALLTHRU */
+	    case 6: z *= (y+(float)5.0);	/* FALLTHRU */
+	    case 5: z *= (y+(float)4.0);	/* FALLTHRU */
+	    case 4: z *= (y+(float)3.0);	/* FALLTHRU */
+	    case 3: z *= (y+(float)2.0);	/* FALLTHRU */
+		    r += __ieee754_logf(z); break;
+	    }
+    /* 8.0 <= x < 2**58 */
+	} else if (ix < 0x5c800000) {
+	    t = __ieee754_logf(x);
+	    z = one/x;
+	    y = z*z;
+	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
+	    r = (x-half)*(t-one)+w;
+	} else 
+    /* 2**58 <= x <= inf */
+	    r =  x*(__ieee754_logf(x)-one);
+	if(hx<0) r = nadj - r;
+	return r;
+}
--- a/lib/libm/math.c
+++ b/lib/libm/math.c
@@ -117,13 +117,6 @@ float log10f(float x) { return logf(x) / (float)_M_LN10; }

 float tanhf(float x) { return sinhf(x) / coshf(x); }

-// TODO we need import these functions from some library (eg musl or newlib)
-float tgammaf(float x) { return 0.0; }
-float lgammaf(float x) { return 0.0; }
-float erff(float x) { return 0.0; }
-float erfcf(float x) { return 0.0; }
-float ldexpf(float x, int exp) { return 0.0; }
-
 /*****************************************************************************/
 /*****************************************************************************/
 // __fpclassifyf from musl-0.9.15

--- a/lib/libm/sf_erf.c
+++ b/lib/libm/sf_erf.c
+/*
+ * This file is part of the Micro Python project, http://micropython.org/
+ *
+ * These math functions are taken from newlib-nano-2, the newlib/libm/math
+ * directory, available from https://github.com/32bitmicro/newlib-nano-2.
+ *
+ * Appropriate copyright headers are reproduced below.
+ */
+
+/* sf_erf.c -- float version of s_erf.c.
+ * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
+ */
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+#include "fdlibm.h"
+
+#define __ieee754_expf expf
+
+#ifdef __v810__
+#define const
+#endif
+
+#ifdef __STDC__
+static const float
+#else
+static float
+#endif
+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] 
+ */
+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 */
+/*
+ * Coefficients for approximation to  erfc in [1.25,1/0.35]
+ */
+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 */
+/*
+ * Coefficients for approximation to  erfc in [1/.35,28]
+ */
+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 */
+
+#ifdef __STDC__
+	float erff(float x) 
+#else
+	float erff(x) 
+	float x;
+#endif
+{
+	__int32_t hx,ix,i;
+	float R,S,P,Q,s,y,z,r;
+	GET_FLOAT_WORD(hx,x);
+	ix = hx&0x7fffffff;
+	if(!FLT_UWORD_IS_FINITE(ix)) {		/* erf(nan)=nan */
+	    i = ((__uint32_t)hx>>31)<<1;
+	    return (float)(1-i)+one/x;	/* erf(+-inf)=+-1 */
+	}
+
+	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);
+		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))));
+	    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)))));
+	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
+	}
+	if (ix >= 0x40c00000) {		/* inf>|x|>=6 */
+	    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)))))));
+	} 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))))));
+	}
+	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);
+	if(hx>=0) return one-r/x; else return  r/x-one;
+}
+
+#ifdef __STDC__
+	float erfcf(float x) 
+#else
+	float erfcf(x) 
+	float x;
+#endif
+{
+	__int32_t hx,ix;
+	float R,S,P,Q,s,y,z,r;
+	GET_FLOAT_WORD(hx,x);
+	ix = hx&0x7fffffff;
+	if(!FLT_UWORD_IS_FINITE(ix)) {			/* erfc(nan)=nan */
+						/* erfc(+-inf)=0,2 */
+	    return (float)(((__uint32_t)hx>>31)<<1)+one/x;
+	}
+
+	if(ix < 0x3f580000) {		/* |x|<0.84375 */
+	    if(ix < 0x23800000)  	/* |x|<2**-56 */
+		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))));
+	    y = r/s;
+	    if(hx < 0x3e800000) {  	/* x<1/4 */
+		return one-(x+x*y);
+	    } else {
+		r = x*y;
+		r += (x-half);
+	        return half - r ;
+	    }
+	}
+	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)))));
+	    if(hx>=0) {
+	        z  = one-erx; return z - P/Q; 
+	    } else {
+		z = erx+P/Q; return one+z;
+	    }
+	}
+	if (ix < 0x41e00000) {		/* |x|<28 */
+	    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)))))));
+	    } 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))))));
+	    }
+	    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);
+	    if(hx>0) return r/x; else return two-r/x;
+	} else {
+	    if(hx>0) return tiny*tiny; else return two-tiny;
+	}
+}
+
+#ifdef _DOUBLE_IS_32BITS
+
+#ifdef __STDC__
+	double erf(double x)
+#else
+	double erf(x)
+	double x;
+#endif
+{
+	return (double) erff((float) x);
+}
+
+#ifdef __STDC__
+	double erfc(double x)
+#else
+	double erfc(x)
+	double x;
+#endif
+{
+	return (double) erfcf((float) x);
+}
+
+#endif /* defined(_DOUBLE_IS_32BITS) */
--- a/lib/libm/sf_ldexp.c
+++ b/lib/libm/sf_ldexp.c
+/*
+ * This file is part of the Micro Python project, http://micropython.org/
+ *
+ * These math functions are taken from newlib-nano-2, the newlib/libm/math
+ * directory, available from https://github.com/32bitmicro/newlib-nano-2.
+ *
+ * Appropriate copyright headers are reproduced below.
+ */
+
+/* sf_ldexp.c -- float version of s_ldexp.c.
+ * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
+ */
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ */
+
+#include "fdlibm.h"
+//#include <errno.h>
+
+#ifdef __STDC__
+	float ldexpf(float value, int exp)
+#else
+	float ldexpf(value, exp)
+	float value; int exp;
+#endif
+{
+	if(!finitef(value)||value==(float)0.0) return value;
+	value = scalbnf(value,exp);
+	//if(!finitef(value)||value==(float)0.0) errno = ERANGE;
+	return value;
+}
+
+#ifdef _DOUBLE_IS_32BITS
+
+#ifdef __STDC__
+	double ldexp(double value, int exp)
+#else
+	double ldexp(value, exp)
+	double value; int exp;
+#endif
+{
+	return (double) ldexpf((float) value, exp);
+}
+
+#endif /* defined(_DOUBLE_IS_32BITS) */
--- a/lib/libm/wf_lgamma.c
+++ b/lib/libm/wf_lgamma.c
+/*
+ * This file is part of the Micro Python project, http://micropython.org/
+ *
+ * These math functions are taken from newlib-nano-2, the newlib/libm/math
+ * directory, available from https://github.com/32bitmicro/newlib-nano-2.
+ *
+ * Appropriate copyright headers are reproduced below.
+ */
+
+/* wf_lgamma.c -- float version of w_lgamma.c.
+ * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
+ */
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice 
+ * is preserved.
+ * ====================================================
+ *