Totally rework the math library, this time based on the MacOs X

math library (which is itself based on the math lib from FreeBSD).
 -Erik

1 files changed, 0 insertions, 848 deletions

diff --git a/libm/float/jvf.c b/libm/float/jvf.c
deleted file mode 100644
index 268a8e4eb..000000000
--- a/libm/float/jvf.c
+++ /dev/null
@@ -1,848 +0,0 @@
-/*							jvf.c
- *
- *	Bessel function of noninteger order
- *
- *
- *
- * SYNOPSIS:
- *
- * float v, x, y, jvf();
- *
- * y = jvf( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order v of the argument,
- * where v is real.  Negative x is allowed if v is an integer.
- *
- * Several expansions are included: the ascending power
- * series, the Hankel expansion, and two transitional
- * expansions for large v.  If v is not too large, it
- * is reduced by recurrence to a region of best accuracy.
- *
- * The single precision routine accepts negative v, but with
- * reduced accuracy.
- *
- *
- *
- * ACCURACY:
- * Results for integer v are indicated by *.
- * Error criterion is absolute, except relative when |jv()| > 1.
- *
- * arithmetic     domain      # trials      peak         rms
- *                v      x
- *    IEEE       0,125  0,125   30000      2.0e-6      2.0e-7
- *    IEEE     -17,0    0,125   30000      1.1e-5      4.0e-7
- *    IEEE    -100,0    0,125    3000      1.5e-4      7.8e-6
- */
-
-
-/*
-Cephes Math Library Release 2.2: June, 1992
-Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
-Direct inquiries to 30 Frost Street, Cambridge, MA 02140
-*/
-
-
-#include <math.h>
-#define DEBUG 0
-
-extern float MAXNUMF, MACHEPF, MINLOGF, MAXLOGF, PIF;
-extern int sgngamf;
-
-/* BIG = 1/MACHEPF */
-#define BIG   16777216.
-
-#ifdef ANSIC
-float floorf(float), j0f(float), j1f(float);
-static float jnxf(float, float);
-static float jvsf(float, float);
-static float hankelf(float, float);
-static float jntf(float, float);
-static float recurf( float *, float, float * );
-float sqrtf(float), sinf(float), cosf(float);
-float lgamf(float), expf(float), logf(float), powf(float, float);
-float gammaf(float), cbrtf(float), acosf(float);
-int airyf(float, float *, float *, float *, float *);
-float polevlf(float, float *, int);
-#else
-float floorf(), j0f(), j1f();
-float sqrtf(), sinf(), cosf();
-float lgamf(), expf(), logf(), powf(), gammaf();
-float cbrtf(), polevlf(), acosf();
-void airyf();
-static float recurf(), jvsf(), hankelf(), jnxf(), jntf(), jvsf();
-#endif
-
-
-#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
-
-float jvf( float nn, float xx )
-{
-float n, x, k, q, t, y, an, sign;
-int i, nint;
-
-n = nn;
-x = xx;
-nint = 0;	/* Flag for integer n */
-sign = 1.0;	/* Flag for sign inversion */
-an = fabsf( n );
-y = floorf( an );
-if( y == an )
-	{
-	nint = 1;
-	i = an - 16384.0 * floorf( an/16384.0 );
-	if( n < 0.0 )
-		{
-		if( i & 1 )
-			sign = -sign;
-		n = an;
-		}
-	if( x < 0.0 )
-		{
-		if( i & 1 )
-			sign = -sign;
-		x = -x;
-		}
-	if( n == 0.0 )
-		return( j0f(x) );
-	if( n == 1.0 )
-		return( sign * j1f(x) );
-	}
-
-if( (x < 0.0) && (y != an) )
-	{
-	mtherr( "jvf", DOMAIN );
-	y = 0.0;
-	goto done;
-	}
-
-y = fabsf(x);
-
-if( y < MACHEPF )
-	goto underf;
-
-/* Easy cases - x small compared to n */
-t = 3.6 * sqrtf(an);
-if( y < t )
-	return( sign * jvsf(n,x) );
-
-/* x large compared to n */
-k = 3.6 * sqrtf(y);
-if( (an < k) && (y > 6.0) )
-	return( sign * hankelf(n,x) );
-
-if( (n > -100) && (n < 14.0) )
-	{
-/* Note: if x is too large, the continued
- * fraction will fail; but then the
- * Hankel expansion can be used.
- */
-	if( nint != 0 )
-		{
-		k = 0.0;
-		q = recurf( &n, x, &k );
-		if( k == 0.0 )
-			{
-			y = j0f(x)/q;
-			goto done;
-			}
-		if( k == 1.0 )
-			{
-			y = j1f(x)/q;
-			goto done;
-			}
-		}
-
-	if( n >= 0.0 )
-		{
-/* Recur backwards from a larger value of n
- */
-		if( y > 1.3 * an )
-			goto recurdwn;
-		if( an > 1.3 * y )
-			goto recurdwn;
-		k = n;
-		y = 2.0*(y+an+1.0);
-		if( (y - n) > 33.0 )
-			y = n + 33.0;
-		y = n + floorf(y-n);
-		q = recurf( &y, x, &k );
-		y = jvsf(y,x) * q;
-		goto done;
-		}
-recurdwn:
-	if( an > (k + 3.0) )
-		{
-/* Recur backwards from n to k
- */
-		if( n < 0.0 )
-			k = -k;
-		q = n - floorf(n);
-		k = floorf(k) + q;
-		if( n > 0.0 )
-			q = recurf( &n, x, &k );
-		else
-			{
-			t = k;
-			k = n;
-			q = recurf( &t, x, &k );
-			k = t;
-			}
-		if( q == 0.0 )
-			{
-underf:
-			y = 0.0;
-			goto done;
-			}
-		}
-	else
-		{
-		k = n;
-		q = 1.0;
-		}
-
-/* boundary between convergence of
- * power series and Hankel expansion
- */
- 	t = fabsf(k);
-	if( t < 26.0 )
-		t = (0.0083*t + 0.09)*t + 12.9;
-	else
-		t = 0.9 * t;
-
-	if( y > t ) /* y = |x| */
-		y = hankelf(k,x);
-	else
-		y = jvsf(k,x);
-#if DEBUG
-printf( "y = %.16e, q = %.16e\n", y, q );
-#endif
-	if( n > 0.0 )
-		y /= q;
-	else
-		y *= q;
-	}
-
-else
-	{
-/* For large positive n, use the uniform expansion
- * or the transitional expansion.
- * But if x is of the order of n**2,
- * these may blow up, whereas the
- * Hankel expansion will then work.
- */
-	if( n < 0.0 )
-		{
-		mtherr( "jvf", TLOSS );
-		y = 0.0;
-		goto done;
-		}
-	t = y/an;
-	t /= an;
-	if( t > 0.3 )
-		y = hankelf(n,x);
-	else
-		y = jnxf(n,x);
-	}
-
-done:	return( sign * y);
-}
-
-/* Reduce the order by backward recurrence.
- * AMS55 #9.1.27 and 9.1.73.
- */
-
-static float recurf( float *n, float xx, float *newn )
-{
-float x, pkm2, pkm1, pk, pkp1, qkm2, qkm1;
-float k, ans, qk, xk, yk, r, t, kf, xinv;
-static float big = BIG;
-int nflag, ctr;
-
-x = xx;
-/* continued fraction for Jn(x)/Jn-1(x)  */
-if( *n < 0.0 )
-	nflag = 1;
-else
-	nflag = 0;
-
-fstart:
-
-#if DEBUG
-printf( "n = %.6e, newn = %.6e, cfrac = ", *n, *newn );
-#endif
-
-pkm2 = 0.0;
-qkm2 = 1.0;
-pkm1 = x;
-qkm1 = *n + *n;
-xk = -x * x;
-yk = qkm1;
-ans = 1.0;
-ctr = 0;
-do
-	{
-	yk += 2.0;
-	pk = pkm1 * yk +  pkm2 * xk;
-	qk = qkm1 * yk +  qkm2 * xk;
-	pkm2 = pkm1;
-	pkm1 = pk;
-	qkm2 = qkm1;
-	qkm1 = qk;
-	if( qk != 0 )
-		r = pk/qk;
-	else
-		r = 0.0;
-	if( r != 0 )
-		{
-		t = fabsf( (ans - r)/r );
-		ans = r;
-		}
-	else
-		t = 1.0;
-
-	if( t < MACHEPF )
-		goto done;
-
-	if( fabsf(pk) > big )
-		{
-		pkm2 *= MACHEPF;
-		pkm1 *= MACHEPF;
-		qkm2 *= MACHEPF;
-		qkm1 *= MACHEPF;
-		}
-	}
-while( t > MACHEPF );
-
-done:
-
-#if DEBUG
-printf( "%.6e\n", ans );
-#endif
-
-/* Change n to n-1 if n < 0 and the continued fraction is small
- */
-if( nflag > 0 )
-	{
-	if( fabsf(ans) < 0.125 )
-		{
-		nflag = -1;
-		*n = *n - 1.0;
-		goto fstart;
-		}
-	}
-
-
-kf = *newn;
-
-/* backward recurrence
- *              2k
- *  J   (x)  =  --- J (x)  -  J   (x)
- *   k-1         x   k         k+1
- */
-
-pk = 1.0;
-pkm1 = 1.0/ans;
-k = *n - 1.0;
-r = 2 * k;
-xinv = 1.0/x;
-do
-	{
-	pkm2 = (pkm1 * r  -  pk * x) * xinv;
-	pkp1 = pk;
-	pk = pkm1;
-	pkm1 = pkm2;
-	r -= 2.0;
-#if 0
-	t = fabsf(pkp1) + fabsf(pk);
-	if( (k > (kf + 2.5)) && (fabsf(pkm1) < 0.25*t) )
-		{
-		k -= 1.0;
-		t = x*x;
-		pkm2 = ( (r*(r+2.0)-t)*pk - r*x*pkp1 )/t;
-		pkp1 = pk;
-		pk = pkm1;
-		pkm1 = pkm2;
-		r -= 2.0;
-		}
-#endif
-	k -= 1.0;
-	}
-while( k > (kf + 0.5) );
-
-#if 0
-/* Take the larger of the last two iterates
- * on the theory that it may have less cancellation error.
- */
-if( (kf >= 0.0) && (fabsf(pk) > fabsf(pkm1)) )
-	{
-	k += 1.0;
-	pkm2 = pk;
-	}
-#endif
-
-*newn = k;
-#if DEBUG
-printf( "newn %.6e\n", k );
-#endif
-return( pkm2 );
-}
-
-
-
-/* Ascending power series for Jv(x).
- * AMS55 #9.1.10.
- */
-
-static float jvsf( float nn, float xx )
-{
-float n, x, t, u, y, z, k, ay;
-
-#if DEBUG
-printf( "jvsf: " );
-#endif
-n = nn;
-x = xx;
-z = -0.25 * x * x;
-u = 1.0;
-y = u;
-k = 1.0;
-t = 1.0;
-
-while( t > MACHEPF )
-	{
-	u *= z / (k * (n+k));
-	y += u;
-	k += 1.0;
-	t = fabsf(u);
-	if( (ay = fabsf(y)) > 1.0 )
-		t /= ay;
-	}
-
-if( x < 0.0 )
-	{
-	y = y * powf( 0.5 * x, n ) / gammaf( n + 1.0 );
-	}
-else
-	{
-	t = n * logf(0.5*x) - lgamf(n + 1.0);
-	if( t < -MAXLOGF )
-		{
-		return( 0.0 );
-		}
-	if( t > MAXLOGF )
-		{
-		t = logf(y) + t;
-		if( t > MAXLOGF )
-			{
-			mtherr( "jvf", OVERFLOW );
-			return( MAXNUMF );
-			}
-		else
-			{
-			y = sgngamf * expf(t);
-			return(y);
-			}
-		}
-	y = sgngamf * y * expf( t );
-	}
-#if DEBUG
-printf( "y = %.8e\n", y );
-#endif
-return(y);
-}
-
-/* Hankel's asymptotic expansion
- * for large x.
- * AMS55 #9.2.5.
- */
-static float hankelf( float nn, float xx )
-{
-float n, x, t, u, z, k, sign, conv;
-float p, q, j, m, pp, qq;
-int flag;
-
-#if DEBUG
-printf( "hankelf: " );
-#endif
-n = nn;
-x = xx;
-m = 4.0*n*n;
-j = 1.0;
-z = 8.0 * x;
-k = 1.0;
-p = 1.0;
-u = (m - 1.0)/z;
-q = u;
-sign = 1.0;
-conv = 1.0;
-flag = 0;
-t = 1.0;
-pp = 1.0e38;
-qq = 1.0e38;
-
-while( t > MACHEPF )
-	{
-	k += 2.0;
-	j += 1.0;
-	sign = -sign;
-	u *= (m - k * k)/(j * z);
-	p += sign * u;
-	k += 2.0;
-	j += 1.0;
-	u *= (m - k * k)/(j * z);
-	q += sign * u;
-	t = fabsf(u/p);
-	if( t < conv )
-		{
-		conv = t;
-		qq = q;
-		pp = p;
-		flag = 1;
-		}
-/* stop if the terms start getting larger */
-	if( (flag != 0) && (t > conv) )
-		{
-#if DEBUG
-		printf( "Hankel: convergence to %.4E\n", conv );
-#endif
-		goto hank1;
-		}
-	}	
-
-hank1:
-u = x - (0.5*n + 0.25) * PIF;
-t = sqrtf( 2.0/(PIF*x) ) * ( pp * cosf(u) - qq * sinf(u) );
-return( t );
-}
-
-
-/* Asymptotic expansion for large n.
- * AMS55 #9.3.35.
- */
-
-static float lambda[] = {
-  1.0,
-  1.041666666666666666666667E-1,
-  8.355034722222222222222222E-2,
-  1.282265745563271604938272E-1,
-  2.918490264641404642489712E-1,
-  8.816272674437576524187671E-1,
-  3.321408281862767544702647E+0,
-  1.499576298686255465867237E+1,
-  7.892301301158651813848139E+1,
-  4.744515388682643231611949E+2,
-  3.207490090890661934704328E+3
-};
-static float mu[] = {
-  1.0,
- -1.458333333333333333333333E-1,
- -9.874131944444444444444444E-2,
- -1.433120539158950617283951E-1,
- -3.172272026784135480967078E-1,
- -9.424291479571202491373028E-1,
- -3.511203040826354261542798E+0,
- -1.572726362036804512982712E+1,
- -8.228143909718594444224656E+1,
- -4.923553705236705240352022E+2,
- -3.316218568547972508762102E+3
-};
-static float P1[] = {
- -2.083333333333333333333333E-1,
-  1.250000000000000000000000E-1
-};
-static float P2[] = {
-  3.342013888888888888888889E-1,
- -4.010416666666666666666667E-1,
-  7.031250000000000000000000E-2
-};
-static float P3[] = {
- -1.025812596450617283950617E+0,
-  1.846462673611111111111111E+0,
- -8.912109375000000000000000E-1,
-  7.324218750000000000000000E-2
-};
-static float P4[] = {
-  4.669584423426247427983539E+0,
- -1.120700261622299382716049E+1,
-  8.789123535156250000000000E+0,
- -2.364086914062500000000000E+0,
-  1.121520996093750000000000E-1
-};
-static float P5[] = {
- -2.8212072558200244877E1,
-  8.4636217674600734632E1,
- -9.1818241543240017361E1,
-  4.2534998745388454861E1,
- -7.3687943594796316964E0,
-  2.27108001708984375E-1
-};
-static float P6[] = {
-  2.1257013003921712286E2,
- -7.6525246814118164230E2,
-  1.0599904525279998779E3,
- -6.9957962737613254123E2,
-  2.1819051174421159048E2,
- -2.6491430486951555525E1,
-  5.7250142097473144531E-1
-};
-static float P7[] = {
- -1.9194576623184069963E3,
-  8.0617221817373093845E3,
- -1.3586550006434137439E4,
-  1.1655393336864533248E4,
- -5.3056469786134031084E3,
-  1.2009029132163524628E3,
- -1.0809091978839465550E2,
-  1.7277275025844573975E0
-};
-
-
-static float jnxf( float nn, float xx )
-{
-float n, x, zeta, sqz, zz, zp, np;
-float cbn, n23, t, z, sz;
-float pp, qq, z32i, zzi;
-float ak, bk, akl, bkl;
-int sign, doa, dob, nflg, k, s, tk, tkp1, m;
-static float u[8];
-static float ai, aip, bi, bip;
-
-n = nn;
-x = xx;
-/* Test for x very close to n.
- * Use expansion for transition region if so.
- */
-cbn = cbrtf(n);
-z = (x - n)/cbn;
-if( (fabsf(z) <= 0.7) || (n < 0.0) )
-	return( jntf(n,x) );
-z = x/n;
-zz = 1.0 - z*z;
-if( zz == 0.0 )
-	return(0.0);
-
-if( zz > 0.0 )
-	{
-	sz = sqrtf( zz );
-	t = 1.5 * (logf( (1.0+sz)/z ) - sz );	/* zeta ** 3/2		*/
-	zeta = cbrtf( t * t );
-	nflg = 1;
-	}
-else
-	{
-	sz = sqrtf(-zz);
-	t = 1.5 * (sz - acosf(1.0/z));
-	zeta = -cbrtf( t * t );
-	nflg = -1;
-	}
-z32i = fabsf(1.0/t);
-sqz = cbrtf(t);
-
-/* Airy function */
-n23 = cbrtf( n * n );
-t = n23 * zeta;
-
-#if DEBUG
-printf("zeta %.5E, Airyf(%.5E)\n", zeta, t );
-#endif
-airyf( t, &ai, &aip, &bi, &bip );
-
-/* polynomials in expansion */
-u[0] = 1.0;
-zzi = 1.0/zz;
-u[1] = polevlf( zzi, P1, 1 )/sz;
-u[2] = polevlf( zzi, P2, 2 )/zz;
-u[3] = polevlf( zzi, P3, 3 )/(sz*zz);
-pp = zz*zz;
-u[4] = polevlf( zzi, P4, 4 )/pp;
-u[5] = polevlf( zzi, P5, 5 )/(pp*sz);
-pp *= zz;
-u[6] = polevlf( zzi, P6, 6 )/pp;
-u[7] = polevlf( zzi, P7, 7 )/(pp*sz);
-
-#if DEBUG
-for( k=0; k<=7; k++ )
-	printf( "u[%d] = %.5E\n", k, u[k] );
-#endif
-
-pp = 0.0;
-qq = 0.0;
-np = 1.0;
-/* flags to stop when terms get larger */
-doa = 1;
-dob = 1;
-akl = MAXNUMF;
-bkl = MAXNUMF;
-
-for( k=0; k<=3; k++ )
-	{
-	tk = 2 * k;
-	tkp1 = tk + 1;
-	zp = 1.0;
-	ak = 0.0;
-	bk = 0.0;
-	for( s=0; s<=tk; s++ )
-		{
-		if( doa )
-			{
-			if( (s & 3) > 1 )
-				sign = nflg;
-			else
-				sign = 1;
-			ak += sign * mu[s] * zp * u[tk-s];
-			}
-
-		if( dob )
-			{
-			m = tkp1 - s;
-			if( ((m+1) & 3) > 1 )
-				sign = nflg;
-			else
-				sign = 1;
-			bk += sign * lambda[s] * zp * u[m];
-			}
-		zp *= z32i;
-		}
-
-	if( doa )
-		{
-		ak *= np;
-		t = fabsf(ak);
-		if( t < akl )
-			{
-			akl = t;
-			pp += ak;
-			}
-		else
-			doa = 0;
-		}
-
-	if( dob )
-		{
-		bk += lambda[tkp1] * zp * u[0];
-		bk *= -np/sqz;
-		t = fabsf(bk);
-		if( t < bkl )
-			{
-			bkl = t;
-			qq += bk;
-			}
-		else
-			dob = 0;
-		}
-#if DEBUG
-	printf("a[%d] %.5E, b[%d] %.5E\n", k, ak, k, bk );
-#endif
-	if( np < MACHEPF )
-		break;
-	np /= n*n;
-	}
-
-/* normalizing factor ( 4*zeta/(1 - z**2) )**1/4	*/
-t = 4.0 * zeta/zz;
-t = sqrtf( sqrtf(t) );
-
-t *= ai*pp/cbrtf(n)  +  aip*qq/(n23*n);
-return(t);
-}
-
-/* Asymptotic expansion for transition region,
- * n large and x close to n.
- * AMS55 #9.3.23.
- */
-
-static float PF2[] = {
- -9.0000000000000000000e-2,
-  8.5714285714285714286e-2
-};
-static float PF3[] = {
-  1.3671428571428571429e-1,
- -5.4920634920634920635e-2,
- -4.4444444444444444444e-3
-};
-static float PF4[] = {
-  1.3500000000000000000e-3,
- -1.6036054421768707483e-1,
-  4.2590187590187590188e-2,
-  2.7330447330447330447e-3
-};
-static float PG1[] = {
- -2.4285714285714285714e-1,
-  1.4285714285714285714e-2
-};
-static float PG2[] = {
- -9.0000000000000000000e-3,
-  1.9396825396825396825e-1,
- -1.1746031746031746032e-2
-};
-static float PG3[] = {
-  1.9607142857142857143e-2,
- -1.5983694083694083694e-1,
-  6.3838383838383838384e-3
-};
-
-
-static float jntf( float nn, float xx )
-{
-float n, x, z, zz, z3;
-float cbn, n23, cbtwo;
-float ai, aip, bi, bip;	/* Airy functions */
-float nk, fk, gk, pp, qq;
-float F[5], G[4];
-int k;
-
-n = nn;
-x = xx;
-cbn = cbrtf(n);
-z = (x - n)/cbn;
-cbtwo = cbrtf( 2.0 );
-
-/* Airy function */
-zz = -cbtwo * z;
-airyf( zz, &ai, &aip, &bi, &bip );
-
-/* polynomials in expansion */
-zz = z * z;
-z3 = zz * z;
-F[0] = 1.0;
-F[1] = -z/5.0;
-F[2] = polevlf( z3, PF2, 1 ) * zz;
-F[3] = polevlf( z3, PF3, 2 );
-F[4] = polevlf( z3, PF4, 3 ) * z;
-G[0] = 0.3 * zz;
-G[1] = polevlf( z3, PG1, 1 );
-G[2] = polevlf( z3, PG2, 2 ) * z;
-G[3] = polevlf( z3, PG3, 2 ) * zz;
-#if DEBUG
-for( k=0; k<=4; k++ )
-	printf( "F[%d] = %.5E\n", k, F[k] );
-for( k=0; k<=3; k++ )
-	printf( "G[%d] = %.5E\n", k, G[k] );
-#endif
-pp = 0.0;
-qq = 0.0;
-nk = 1.0;
-n23 = cbrtf( n * n );
-
-for( k=0; k<=4; k++ )
-	{
-	fk = F[k]*nk;
-	pp += fk;
-	if( k != 4 )
-		{
-		gk = G[k]*nk;
-		qq += gk;
-		}
-#if DEBUG
-	printf("fk[%d] %.5E, gk[%d] %.5E\n", k, fk, k, gk );
-#endif
-	nk /= n23;
-	}
-
-fk = cbtwo * ai * pp/cbn  +  cbrtf(4.0) * aip * qq/n;
-return(fk);
-}