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authorSzabolcs Nagy <nsz@port70.net>2013-01-01 22:11:28 +0100
committerSzabolcs Nagy <nsz@port70.net>2013-01-01 22:11:28 +0100
commit5bb6b24952e3f95ede42b60ac64a33ac34b8e272 (patch)
tree26a9f099e22ab2e3f3675aac619db3b093de551f
parent697acde67e0da4d73b46445ed536fe9923d515c7 (diff)
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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)
-rw-r--r--src/math/j1.c172
-rw-r--r--src/math/j1f.c153
2 files changed, 138 insertions, 187 deletions
diff --git a/src/math/j1.c b/src/math/j1.c
index 4a2cba8d..ac7bb1eb 100644
--- a/src/math/j1.c
+++ b/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])))));
diff --git a/src/math/j1f.c b/src/math/j1f.c
index de459e0d..5a760f71 100644
--- a/src/math/j1f.c
+++ b/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])))));