math: bessel cleanup (j1.c and j1f.c)

a common code path in j1 and y1 was factored out so the resulting
object code is a bit smaller

unsigned int arithmetics is used for bit manipulation

j1(-inf) now returns 0 instead of -0

an incorrect threshold in the common code of j1f and y1f got fixed
(this caused spurious overflow and underflow exceptions)

the else branch in pone and pzero functions are fixed
(so code analyzers dont warn about uninitialized values)

src/math/j1.c

@@ -59,10 +59,47 @@
 static double pone(double), qone(double);
 
 static const double
-huge      = 1e300,
 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
-tpi       = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
+tpi       = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
+
+static double common(uint32_t ix, double x, int y1, int sign)
+{
+	double z,s,c,ss,cc;
+
+	/*
+	 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
+	 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
+	 *
+	 * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
+	 * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
+	 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+	 */
+	s = sin(x);
+	if (y1)
+		s = -s;
+	c = cos(x);
+	cc = s-c;
+	if (ix < 0x7fe00000) {
+		/* avoid overflow in 2*x */
+		ss = -s-c;
+		z = cos(2*x);
+		if (s*c > 0)
+			cc = z/ss;
+		else
+			ss = z/cc;
+		if (ix < 0x48000000) {
+			if (y1)
+				ss = -ss;
+			cc = pone(x)*cc-qone(x)*ss;
+		}
+	}
+	if (sign)
+		cc = -cc;
+	return invsqrtpi*cc/sqrt(x);
+}
+
 /* R0/S0 on [0,2] */
+static const double
 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
 r01 =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
@@ -75,52 +112,26 @@ s05 =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
 
 double j1(double x)
 {
-	double z,s,c,ss,cc,r,u,v,y;
-	int32_t hx,ix;
+	double z,r,s;
+	uint32_t ix;
+	int sign;
 
-	GET_HIGH_WORD(hx, x);
-	ix = hx & 0x7fffffff;
+	GET_HIGH_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
 	if (ix >= 0x7ff00000)
-		return 1.0/x;
-	y = fabs(x);
-	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
-		s = sin(y);
-		c = cos(y);
-		ss = -s-c;
-		cc = s-c;
-		if (ix < 0x7fe00000) {  /* make sure y+y not overflow */
-			z = cos(y+y);
-			if (s*c > 0.0)
-				cc = z/ss;
-			else
-				ss = z/cc;
-		}
-		/*
-		 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
-		 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
-		 */
-		if (ix > 0x48000000)
-			z = (invsqrtpi*cc)/sqrt(y);
-		else {
-			u = pone(y);
-			v = qone(y);
-			z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
-		}
-		if (hx < 0)
-			return -z;
-		else
-			return  z;
-	}
-	if (ix < 0x3e400000) {  /* |x| < 2**-27 */
-		/* raise inexact if x!=0 */
-		if (huge+x > 1.0)
-			return 0.5*x;
-	}
-	z = x*x;
-	r = z*(r00+z*(r01+z*(r02+z*r03)));
-	s = 1.0+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
-	r *= x;
-	return x*0.5 + r/s;
+		return 1/(x*x);
+	if (ix >= 0x40000000)  /* |x| >= 2 */
+		return common(ix, fabs(x), 0, sign);
+	if (ix >= 0x38000000) {  /* |x| >= 2**-127 */
+		z = x*x;
+		r = z*(r00+z*(r01+z*(r02+z*r03)));
+		s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
+		z = r/s;
+	} else
+		/* avoid underflow, raise inexact if x!=0 */
+		z = x;
+	return (0.5 + z)*x;
 }
 
 static const double U0[5] = {
@@ -138,59 +149,28 @@ static const double V0[5] = {
   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
 };
 
-
 double y1(double x)
 {
-	double z,s,c,ss,cc,u,v;
-	int32_t hx,ix,lx;
+	double z,u,v;
+	uint32_t ix,lx;
 
-	EXTRACT_WORDS(hx, lx, x);
-	ix = 0x7fffffff & hx;
-	/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
+	EXTRACT_WORDS(ix, lx, x);
+	/* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
+	if ((ix<<1 | lx) == 0)
+		return -1/0.0;
+	if (ix>>31)
+		return 0/0.0;
 	if (ix >= 0x7ff00000)
-		return 1.0/(x+x*x);
-	if ((ix|lx) == 0)
-		return -1.0/0.0;
-	if (hx < 0)
-		return 0.0/0.0;
-	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
-		s = sin(x);
-		c = cos(x);
-		ss = -s-c;
-		cc = s-c;
-		if (ix < 0x7fe00000) {  /* make sure x+x not overflow */
-			z = cos(x+x);
-			if (s*c > 0.0)
-				cc = z/ss;
-			else
-				ss = z/cc;
-		}
-		/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
-		 * where x0 = x-3pi/4
-		 *      Better formula:
-		 *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
-		 *                      =  1/sqrt(2) * (sin(x) - cos(x))
-		 *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
-		 *                      = -1/sqrt(2) * (cos(x) + sin(x))
-		 * To avoid cancellation, use
-		 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
-		 * to compute the worse one.
-		 */
-		if (ix > 0x48000000)
-			z = (invsqrtpi*ss)/sqrt(x);
-		else {
-			u = pone(x);
-			v = qone(x);
-			z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
-		}
-		return z;
-	}
-	if (ix <= 0x3c900000)  /* x < 2**-54 */
+		return 1/x;
+
+	if (ix >= 0x40000000)  /* x >= 2 */
+		return common(ix, x, 1, 0);
+	if (ix < 0x3c900000)  /* x < 2**-54 */
 		return -tpi/x;
 	z = x*x;
 	u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
-	v = 1.0+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
-	return x*(u/v) + tpi*(j1(x)*log(x)-1.0/x);
+	v = 1+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
+	return x*(u/v) + tpi*(j1(x)*log(x)-1/x);
 }
 
 /* For x >= 8, the asymptotic expansions of pone is
@@ -271,14 +251,14 @@ static double pone(double x)
 {
 	const double *p,*q;
 	double z,r,s;
-	int32_t ix;
+	uint32_t ix;
 
 	GET_HIGH_WORD(ix, x);
 	ix &= 0x7fffffff;
 	if      (ix >= 0x40200000){p = pr8; q = ps8;}
 	else if (ix >= 0x40122E8B){p = pr5; q = ps5;}
 	else if (ix >= 0x4006DB6D){p = pr3; q = ps3;}
-	else if (ix >= 0x40000000){p = pr2; q = ps2;}
+	else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
 	z = 1.0/(x*x);
 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 	s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
@@ -367,14 +347,14 @@ static double qone(double x)
 {
 	const double *p,*q;
 	double  s,r,z;
-	int32_t ix;
+	uint32_t ix;
 
 	GET_HIGH_WORD(ix, x);
 	ix &= 0x7fffffff;
 	if      (ix >= 0x40200000){p = qr8; q = qs8;}
 	else if (ix >= 0x40122E8B){p = qr5; q = qs5;}
 	else if (ix >= 0x4006DB6D){p = qr3; q = qs3;}
-	else if (ix >= 0x40000000){p = qr2; q = qs2;}
+	else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
 	z = 1.0/(x*x);
 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 	s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));

src/math/j1f.c

@@ -18,10 +18,38 @@
 static float ponef(float), qonef(float);
 
 static const float
-huge      = 1e30,
 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
-tpi       = 6.3661974669e-01, /* 0x3f22f983 */
+tpi       = 6.3661974669e-01; /* 0x3f22f983 */
+
+static float common(uint32_t ix, float x, int y1, int sign)
+{
+	double z,s,c,ss,cc;
+
+	s = sinf(x);
+	if (y1)
+		s = -s;
+	c = cosf(x);
+	cc = s-c;
+	if (ix < 0x7f000000) {
+		ss = -s-c;
+		z = cosf(2*x);
+		if (s*c > 0)
+			cc = z/ss;
+		else
+			ss = z/cc;
+		if (ix < 0x58800000) {
+			if (y1)
+				ss = -ss;
+			cc = ponef(x)*cc-qonef(x)*ss;
+		}
+	}
+	if (sign)
+		cc = -cc;
+	return invsqrtpi*cc/sqrtf(x);
+}
+
 /* R0/S0 on [0,2] */
+static const float
 r00 = -6.2500000000e-02, /* 0xbd800000 */
 r01 =  1.4070566976e-03, /* 0x3ab86cfd */
 r02 = -1.5995563444e-05, /* 0xb7862e36 */
@@ -34,51 +62,26 @@ s05 =  1.2354227016e-11; /* 0x2d59567e */
 
 float j1f(float x)
 {
-	float z,s,c,ss,cc,r,u,v,y;
-	int32_t hx,ix;
+	float z,r,s;
+	uint32_t ix;
+	int sign;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
+	GET_FLOAT_WORD(ix, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
 	if (ix >= 0x7f800000)
-		return 1.0f/x;
-	y = fabsf(x);
-	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
-		s = sinf(y);
-		c = cosf(y);
-		ss = -s-c;
-		cc = s-c;
-		if (ix < 0x7f000000) {  /* make sure y+y not overflow */
-			z = cosf(y+y);
-			if (s*c > 0.0f)
-				cc = z/ss;
-			else
-				ss = z/cc;
-		}
-		/*
-		 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
-		 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
-		 */
-		if (ix > 0x80000000)
-			z = (invsqrtpi*cc)/sqrtf(y);
-		else {
-			u = ponef(y);
-			v = qonef(y);
-			z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
-		}
-		if (hx < 0)
-			return -z;
-		return  z;
-	}
-	if (ix < 0x32000000) {  /* |x| < 2**-27 */
+		return 1/(x*x);
+	if (ix >= 0x40000000)  /* |x| >= 2 */
+		return common(ix, fabsf(x), 0, sign);
+	if (ix >= 0x32000000) {  /* |x| >= 2**-27 */
+		z = x*x;
+		r = z*(r00+z*(r01+z*(r02+z*r03)));
+		s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
+		z = 0.5f + r/s;
+	} else
 		/* raise inexact if x!=0 */
-		if (huge+x > 1.0f)
-			return 0.5f*x;
-	}
-	z = x*x;
-	r = z*(r00+z*(r01+z*(r02+z*r03)));
-	s = 1.0f+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
-	r *= x;
-	return 0.5f*x + r/s;
+		z = 0.5f + x;
+	return z*x;
 }
 
 static const float U0[5] = {
@@ -98,51 +101,19 @@ static const float V0[5] = {
 
 float y1f(float x)
 {
-	float z,s,c,ss,cc,u,v;
-	int32_t hx,ix;
+	float z,u,v;
+	uint32_t ix;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = 0x7fffffff & hx;
-	/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
+	GET_FLOAT_WORD(ix, x);
+	if ((ix & 0x7fffffff) == 0)
+		return -1/0.0f;
+	if (ix>>31)
+		return 0/0.0f;
 	if (ix >= 0x7f800000)
-		return 1.0f/(x+x*x);
-	if (ix == 0)
-		return -1.0f/0.0f;
-	if (hx < 0)
-		return 0.0f/0.0f;
-	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
-		s = sinf(x);
-		c = cosf(x);
-		ss = -s-c;
-		cc = s-c;
-		if (ix < 0x7f000000) {  /* make sure x+x not overflow */
-			z = cosf(x+x);
-			if (s*c > 0.0f)
-				cc = z/ss;
-			else
-				ss = z/cc;
-		}
-		/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
-		 * where x0 = x-3pi/4
-		 *      Better formula:
-		 *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
-		 *                      =  1/sqrt(2) * (sin(x) - cos(x))
-		 *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
-		 *                      = -1/sqrt(2) * (cos(x) + sin(x))
-		 * To avoid cancellation, use
-		 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
-		 * to compute the worse one.
-		 */
-		if (ix > 0x48000000)
-			z = (invsqrtpi*ss)/sqrtf(x);
-		else {
-			u = ponef(x);
-			v = qonef(x);
-			z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
-		}
-		return z;
-	}
-	if (ix <= 0x24800000)  /* x < 2**-54 */
+		return 1/x;
+	if (ix >= 0x40000000)  /* |x| >= 2.0 */
+		return common(ix,x,1,0);
+	if (ix < 0x32000000)  /* x < 2**-27 */
 		return -tpi/x;
 	z = x*x;
 	u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
@@ -228,14 +199,14 @@ static float ponef(float x)
 {
 	const float *p,*q;
 	float z,r,s;
-	int32_t ix;
+	uint32_t ix;
 
 	GET_FLOAT_WORD(ix, x);
 	ix &= 0x7fffffff;
 	if      (ix >= 0x41000000){p = pr8; q = ps8;}
 	else if (ix >= 0x40f71c58){p = pr5; q = ps5;}
 	else if (ix >= 0x4036db68){p = pr3; q = ps3;}
-	else if (ix >= 0x40000000){p = pr2; q = ps2;}
+	else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
 	z = 1.0f/(x*x);
 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 	s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
@@ -324,14 +295,14 @@ static float qonef(float x)
 {
 	const float *p,*q;
 	float s,r,z;
-	int32_t ix;
+	uint32_t ix;
 
 	GET_FLOAT_WORD(ix, x);
 	ix &= 0x7fffffff;
 	if      (ix >= 0x40200000){p = qr8; q = qs8;}
 	else if (ix >= 0x40f71c58){p = qr5; q = qs5;}
 	else if (ix >= 0x4036db68){p = qr3; q = qs3;}
-	else if (ix >= 0x40000000){p = qr2; q = qs2;}
+	else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
 	z = 1.0f/(x*x);
 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 	s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));