#ifndef lint
static char *RCSid = "$Id: standard.c,v 3.26 92/03/24 22:34:37 woo Exp Locker: woo $";
#endif
/* GNUPLOT - standard.c */
/*
* Copyright (C) 1986, 1987, 1990, 1991, 1992 Thomas Williams, Colin Kelley
*
* Permission to use, copy, and distribute this software and its
* documentation for any purpose with or without fee is hereby granted,
* provided that the above copyright notice appear in all copies and
* that both that copyright notice and this permission notice appear
* in supporting documentation.
*
* Permission to modify the software is granted, but not the right to
* distribute the modified code. Modifications are to be distributed
* as patches to released version.
*
* This software is provided "as is" without express or implied warranty.
*
*
* AUTHORS
*
* Original Software:
* Thomas Williams, Colin Kelley.
*
* Gnuplot 2.0 additions:
* Russell Lang, Dave Kotz, John Campbell.
*
* Gnuplot 3.0 additions:
* Gershon Elber and many others.
*
* Send your comments or suggestions to
* info-gnuplot@ames.arc.nasa.gov.
* This is a mailing list; to join it send a note to
* info-gnuplot-request@ames.arc.nasa.gov.
* Send bug reports to
* bug-gnuplot@ames.arc.nasa.gov.
*/
#include <math.h>
#include <stdio.h>
#include "plot.h"
#ifdef vms
#include <errno.h>
#else
extern int errno;
#endif /* vms */
extern struct value stack[STACK_DEPTH];
extern int s_p;
extern double zero;
struct value *pop(), *complex(), *integer();
double magnitude(), angle(), real(), imag();
/* The bessel function approximations here are from
* "Computer Approximations"
* by Hart, Cheney et al.
* John Wiley & Sons, 1968
*/
/* There appears to be a mistake in Hart, Cheney et al. on page 149.
* Where it list Qn(x)/x ~ P(z*z)/Q(z*z), z = 8/x, it should read
* Qn(x)/z ~ P(z*z)/Q(z*z), z = 8/x
* In the functions below, Qn(x) is implementated using the later
* equation.
* These bessel functions are accurate to about 1e-13
*/
#define PI_ON_FOUR 0.78539816339744830961566084581987572
#define PI_ON_TWO 1.57079632679489661923131269163975144
#define THREE_PI_ON_FOUR 2.35619449019234492884698253745962716
#define TWO_ON_PI 0.63661977236758134307553505349005744
static double dzero = 0.0;
/* jzero for x in [0,8]
* Index 5849, 19.22 digits precision
*/
static double pjzero[] = {
0.4933787251794133561816813446e+21,
-0.11791576291076105360384408e+21,
0.6382059341072356562289432465e+19,
-0.1367620353088171386865416609e+18,
0.1434354939140346111664316553e+16,
-0.8085222034853793871199468171e+13,
0.2507158285536881945555156435e+11,
-0.4050412371833132706360663322e+8,
0.2685786856980014981415848441e+5
};
static double qjzero[] = {
0.4933787251794133562113278438e+21,
0.5428918384092285160200195092e+19,
0.3024635616709462698627330784e+17,
0.1127756739679798507056031594e+15,
0.3123043114941213172572469442e+12,
0.669998767298223967181402866e+9,
0.1114636098462985378182402543e+7,
0.1363063652328970604442810507e+4,
0.1e+1
};
/* pzero for x in [8,inf]
* Index 6548, 18.16 digits precision
*/
static double ppzero[] = {
0.2277909019730468430227002627e+5,
0.4134538663958076579678016384e+5,
0.2117052338086494432193395727e+5,
0.348064864432492703474453111e+4,
0.15376201909008354295771715e+3,
0.889615484242104552360748e+0
};
static double qpzero[] = {
0.2277909019730468431768423768e+5,
0.4137041249551041663989198384e+5,
0.2121535056188011573042256764e+5,
0.350287351382356082073561423e+4,
0.15711159858080893649068482e+3,
0.1e+1
};
/* qzero for x in [8,inf]
* Index 6948, 18.33 digits precision
*/
static double pqzero[] = {
-0.8922660020080009409846916e+2,
-0.18591953644342993800252169e+3,
-0.11183429920482737611262123e+3,
-0.2230026166621419847169915e+2,
-0.124410267458356384591379e+1,
-0.8803330304868075181663e-2,
};
static double qqzero[] = {
0.571050241285120619052476459e+4,
0.1195113154343461364695265329e+5,
0.726427801692110188369134506e+4,
0.148872312322837565816134698e+4,
0.9059376959499312585881878e+2,
0.1e+1
};
/* yzero for x in [0,8]
* Index 6245, 18.78 digits precision
*/
static double pyzero[] = {
-0.2750286678629109583701933175e+20,
0.6587473275719554925999402049e+20,
-0.5247065581112764941297350814e+19,
0.1375624316399344078571335453e+18,
-0.1648605817185729473122082537e+16,
0.1025520859686394284509167421e+14,
-0.3436371222979040378171030138e+11,
0.5915213465686889654273830069e+8,
-0.4137035497933148554125235152e+5
};
static double qyzero[] = {
0.3726458838986165881989980739e+21,
0.4192417043410839973904769661e+19,
0.2392883043499781857439356652e+17,
0.9162038034075185262489147968e+14,
0.2613065755041081249568482092e+12,
0.5795122640700729537380087915e+9,
0.1001702641288906265666651753e+7,
0.1282452772478993804176329391e+4,
0.1e+1
};
/* jone for x in [0,8]
* Index 6050, 20.98 digits precision
*/
static double pjone[] = {
0.581199354001606143928050809e+21,
-0.6672106568924916298020941484e+20,
0.2316433580634002297931815435e+19,
-0.3588817569910106050743641413e+17,
0.2908795263834775409737601689e+15,
-0.1322983480332126453125473247e+13,
0.3413234182301700539091292655e+10,
-0.4695753530642995859767162166e+7,
0.270112271089232341485679099e+4
};
static double qjone[] = {
0.11623987080032122878585294e+22,
0.1185770712190320999837113348e+20,
0.6092061398917521746105196863e+17,
0.2081661221307607351240184229e+15,
0.5243710262167649715406728642e+12,
0.1013863514358673989967045588e+10,
0.1501793594998585505921097578e+7,
0.1606931573481487801970916749e+4,
0.1e+1
};
/* pone for x in [8,inf]
* Index 6749, 18.11 digits precision
*/
static double ppone[] = {
0.352246649133679798341724373e+5,
0.62758845247161281269005675e+5,
0.313539631109159574238669888e+5,
0.49854832060594338434500455e+4,
0.2111529182853962382105718e+3,
0.12571716929145341558495e+1
};
static double qpone[] = {
0.352246649133679798068390431e+5,
0.626943469593560511888833731e+5,
0.312404063819041039923015703e+5,
0.4930396490181088979386097e+4,
0.2030775189134759322293574e+3,
0.1e+1
};
/* qone for x in [8,inf]
* Index 7149, 18.28 digits precision
*/
static double pqone[] = {
0.3511751914303552822533318e+3,
0.7210391804904475039280863e+3,
0.4259873011654442389886993e+3,
0.831898957673850827325226e+2,
0.45681716295512267064405e+1,
0.3532840052740123642735e-1
};
static double qqone[] = {
0.74917374171809127714519505e+4,
0.154141773392650970499848051e+5,
0.91522317015169922705904727e+4,
0.18111867005523513506724158e+4,
0.1038187585462133728776636e+3,
0.1e+1
};
/* yone for x in [0,8]
* Index 6444, 18.24 digits precision
*/
static double pyone[] = {
-0.2923821961532962543101048748e+20,
0.7748520682186839645088094202e+19,
-0.3441048063084114446185461344e+18,
0.5915160760490070618496315281e+16,
-0.4863316942567175074828129117e+14,
0.2049696673745662182619800495e+12,
-0.4289471968855248801821819588e+9,
0.3556924009830526056691325215e+6
};
static double qyone[] = {
0.1491311511302920350174081355e+21,
0.1818662841706134986885065935e+19,
0.113163938269888452690508283e+17,
0.4755173588888137713092774006e+14,
0.1500221699156708987166369115e+12,
0.3716660798621930285596927703e+9,
0.726914730719888456980191315e+6,
0.10726961437789255233221267e+4,
0.1e+1
};
f_real()
{
struct value a;
push( complex(&a,real(pop(&a)), 0.0) );
}
f_imag()
{
struct value a;
push( complex(&a,imag(pop(&a)), 0.0) );
}
f_arg()
{
struct value a;
push( complex(&a,angle(pop(&a)), 0.0) );
}
f_conjg()
{
struct value a;
(void) pop(&a);
push( complex(&a,real(&a),-imag(&a) ));
}
f_sin()
{
struct value a;
(void) pop(&a);
push( complex(&a,sin(real(&a))*cosh(imag(&a)), cos(real(&a))*sinh(imag(&a))) );
}
f_cos()
{
struct value a;
(void) pop(&a);
push( complex(&a,cos(real(&a))*cosh(imag(&a)), -sin(real(&a))*sinh(imag(&a))));
}
f_tan()
{
struct value a;
register double den;
(void) pop(&a);
if (imag(&a) == 0.0)
push( complex(&a,tan(real(&a)),0.0) );
else {
den = cos(2*real(&a))+cosh(2*imag(&a));
if (den == 0.0) {
undefined = TRUE;
push( &a );
}
else
push( complex(&a,sin(2*real(&a))/den, sinh(2*imag(&a))/den) );
}
}
f_asin()
{
struct value a;
register double alpha, beta, x, y;
(void) pop(&a);
x = real(&a); y = imag(&a);
if (y == 0.0) {
if (fabs(x) > 1.0) {
undefined = TRUE;
push(complex(&a,0.0, 0.0));
} else
push( complex(&a,asin(x),0.0) );
} else {
beta = sqrt((x + 1)*(x + 1) + y*y)/2 - sqrt((x - 1)*(x - 1) + y*y)/2;
alpha = sqrt((x + 1)*(x + 1) + y*y)/2 + sqrt((x - 1)*(x - 1) + y*y)/2;
push( complex(&a,asin(beta), log(alpha + sqrt(alpha*alpha-1))) );
}
}
f_acos()
{
struct value a;
register double alpha, beta, x, y;
(void) pop(&a);
x = real(&a); y = imag(&a);
if (y == 0.0) {
if (fabs(x) > 1.0) {
undefined = TRUE;
push(complex(&a,0.0, 0.0));
} else
push( complex(&a,acos(x),0.0) );
} else {
alpha = sqrt((x + 1)*(x + 1) + y*y)/2 + sqrt((x - 1)*(x - 1) + y*y)/2;
beta = sqrt((x + 1)*(x + 1) + y*y)/2 - sqrt((x - 1)*(x - 1) + y*y)/2;
push( complex(&a,acos(beta), log(alpha + sqrt(alpha*alpha-1))) );
}
}
f_atan()
{
struct value a;
register double x, y, u, v, w, z;
(void) pop(&a);
x = real(&a); y = imag(&a);
if (y == 0.0)
push( complex(&a,atan(x), 0.0) );
else if (x == 0.0 && fabs(y) == 1.0) {
undefined = TRUE;
push(complex(&a,0.0, 0.0));
} else {
if (x >= 0) {
u = x;
v = y;
} else {
u = -x;
v = -y;
}
z = atan(2*u/(1-u*u-v*v));
w = log((u*u+(v+1)*(v+1))/(u*u+(v-1)*(v-1)))/4;
if (z < 0)
z = z + 2*PI_ON_TWO;
if (x < 0) {
z = -z;
w = -w;
}
push( complex(&a,0.5*z, w) );
}
}
f_sinh()
{
struct value a;
(void) pop(&a);
push( complex(&a,sinh(real(&a))*cos(imag(&a)), cosh(real(&a))*sin(imag(&a))) );
}
f_cosh()
{
struct value a;
(void) pop(&a);
push( complex(&a,cosh(real(&a))*cos(imag(&a)), sinh(real(&a))*sin(imag(&a))) );
}
f_tanh()
{
struct value a;
register double den;
(void) pop(&a);
den = cosh(2*real(&a)) + cos(2*imag(&a));
push( complex(&a,sinh(2*real(&a))/den, sin(2*imag(&a))/den) );
}
f_int()
{
struct value a;
push( integer(&a,(int)real(pop(&a))) );
}
f_abs()
{
struct value a;
(void) pop(&a);
switch (a.type) {
case INT:
push( integer(&a,abs(a.v.int_val)) );
break;
case CMPLX:
push( complex(&a,magnitude(&a), 0.0) );
}
}
f_sgn()
{
struct value a;
(void) pop(&a);
switch(a.type) {
case INT:
push( integer(&a,(a.v.int_val > 0) ? 1 :
(a.v.int_val < 0) ? -1 : 0) );
break;
case CMPLX:
push( integer(&a,(a.v.cmplx_val.real > 0.0) ? 1 :
(a.v.cmplx_val.real < 0.0) ? -1 : 0) );
break;
}
}
f_sqrt()
{
struct value a;
register double mag, ang;
(void) pop(&a);
mag = sqrt(magnitude(&a));
if (imag(&a) == 0.0 && real(&a) < 0.0)
push( complex(&a,0.0,mag) );
else
{
if ( (ang = angle(&a)) < 0.0)
ang += 2*Pi;
ang /= 2;
push( complex(&a,mag*cos(ang), mag*sin(ang)) );
}
}
f_exp()
{
struct value a;
register double mag, ang;
(void) pop(&a);
mag = exp(real(&a));
ang = imag(&a);
push( complex(&a,mag*cos(ang), mag*sin(ang)) );
}
f_log10()
{
struct value a;
register double l10;;
(void) pop(&a);
l10 = log(10.0); /***** replace with a constant! ******/
push( complex(&a,log(magnitude(&a))/l10, angle(&a)/l10) );
}
f_log()
{
struct value a;
(void) pop(&a);
push( complex(&a,log(magnitude(&a)), angle(&a)) );
}
f_floor()
{
struct value a;
(void) pop(&a);
switch (a.type) {
case INT:
push( integer(&a,(int)floor((double)a.v.int_val)));
break;
case CMPLX:
push( integer(&a,(int)floor(a.v.cmplx_val.real)));
}
}
f_ceil()
{
struct value a;
(void) pop(&a);
switch (a.type) {
case INT:
push( integer(&a,(int)ceil((double)a.v.int_val)));
break;
case CMPLX:
push( integer(&a,(int)ceil(a.v.cmplx_val.real)));
}
}
#ifdef GAMMA
f_gamma()
{
extern int signgam;
register double y;
struct value a;
y = GAMMA(real(pop(&a)));
if (y > 88.0) {
undefined = TRUE;
push( integer(&a,0) );
}
else
push( complex(&a,signgam * exp(y),0.0) );
}
#endif /* GAMMA */
/* bessel function approximations */
double jzero(x)
double x;
{
double p, q, x2;
int n;
x2 = x * x;
p = pjzero[8];
q = qjzero[8];
for (n=7; n>=0; n--) {
p = p*x2 + pjzero[n];
q = q*x2 + qjzero[n];
}
return(p/q);
}
double pzero(x)
double x;
{
double p, q, z, z2;
int n;
z = 8.0 / x;
z2 = z * z;
p = ppzero[5];
q = qpzero[5];
for (n=4; n>=0; n--) {
p = p*z2 + ppzero[n];
q = q*z2 + qpzero[n];
}
return(p/q);
}
double qzero(x)
double x;
{
double p, q, z, z2;
int n;
z = 8.0 / x;
z2 = z * z;
p = pqzero[5];
q = qqzero[5];
for (n=4; n>=0; n--) {
p = p*z2 + pqzero[n];
q = q*z2 + qqzero[n];
}
return(p/q);
}
double yzero(x)
double x;
{
double p, q, x2;
int n;
x2 = x * x;
p = pyzero[8];
q = qyzero[8];
for (n=7; n>=0; n--) {
p = p*x2 + pyzero[n];
q = q*x2 + qyzero[n];
}
return(p/q);
}
double rj0(x)
double x;
{
if ( x <= 0.0 )
x = -x;
if ( x < 8.0 )
return(jzero(x));
else
return( sqrt(TWO_ON_PI/x) *
(pzero(x)*cos(x-PI_ON_FOUR) - 8.0/x*qzero(x)*sin(x-PI_ON_FOUR)) );
}
double ry0(x)
double x;
{
if ( x < 0.0 )
return(dzero/dzero); /* error */
if ( x < 8.0 )
return( yzero(x) + TWO_ON_PI*rj0(x)*log(x) );
else
return( sqrt(TWO_ON_PI/x) *
(pzero(x)*sin(x-PI_ON_FOUR) +
(8.0/x)*qzero(x)*cos(x-PI_ON_FOUR)) );
}
double jone(x)
double x;
{
double p, q, x2;
int n;
x2 = x * x;
p = pjone[8];
q = qjone[8];
for (n=7; n>=0; n--) {
p = p*x2 + pjone[n];
q = q*x2 + qjone[n];
}
return(p/q);
}
double pone(x)
double x;
{
double p, q, z, z2;
int n;
z = 8.0 / x;
z2 = z * z;
p = ppone[5];
q = qpone[5];
for (n=4; n>=0; n--) {
p = p*z2 + ppone[n];
q = q*z2 + qpone[n];
}
return(p/q);
}
double qone(x)
double x;
{
double p, q, z, z2;
int n;
z = 8.0 / x;
z2 = z * z;
p = pqone[5];
q = qqone[5];
for (n=4; n>=0; n--) {
p = p*z2 + pqone[n];
q = q*z2 + qqone[n];
}
return(p/q);
}
double yone(x)
double x;
{
double p, q, x2;
int n;
x2 = x * x;
p = 0.0;
q = qyone[8];
for (n=7; n>=0; n--) {
p = p*x2 + pyone[n];
q = q*x2 + qyone[n];
}
return(p/q);
}
double rj1(x)
double x;
{
double v,w;
v = x;
if ( x < 0.0 )
x = -x;
if ( x < 8.0 )
return(v*jone(x));
else {
w = sqrt(TWO_ON_PI/x) *
(pone(x)*cos(x-THREE_PI_ON_FOUR) -
8.0/x*qone(x)*sin(x-THREE_PI_ON_FOUR)) ;
if (v < 0.0)
w = -w;
return( w );
}
}
double ry1(x)
double x;
{
if ( x <= 0.0 )
return(dzero/dzero); /* error */
if ( x < 8.0 )
return( x*yone(x) + TWO_ON_PI*(rj1(x)*log(x) - 1.0/x) );
else
return( sqrt(TWO_ON_PI/x) *
(pone(x)*sin(x-THREE_PI_ON_FOUR) +
(8.0/x)*qone(x)*cos(x-THREE_PI_ON_FOUR)) );
}
f_besj0()
{
struct value a;
double x;
(void) pop(&a);
if (imag(&a) > zero)
int_error("can only do bessel functions of reals",NO_CARET);
push( complex(&a,rj0(real(&a)),0.0) );
}
f_besj1()
{
struct value a;
double x;
(void) pop(&a);
if (imag(&a) > zero)
int_error("can only do bessel functions of reals",NO_CARET);
push( complex(&a,rj1(real(&a)),0.0) );
}
f_besy0()
{
struct value a;
double x;
(void) pop(&a);
if (imag(&a) > zero)
int_error("can only do bessel functions of reals",NO_CARET);
if (real(&a) > 0.0)
push( complex(&a,ry0(real(&a)),0.0) );
else {
push( complex(&a,0.0,0.0) );
undefined = TRUE ;
}
}
f_besy1()
{
struct value a;
double x;
(void) pop(&a);
if (imag(&a) > zero)
int_error("can only do bessel functions of reals",NO_CARET);
if (real(&a) > 0.0)
push( complex(&a,ry1(real(&a)),0.0) );
else {
push( complex(&a,0.0,0.0) );
undefined = TRUE ;
}
}