#ifndef lint
static char *RCSid() { return RCSid("$Id: standard.c,v 1.33 2015/01/20 02:10:43 sfeam Exp $"); }
#endif
/* GNUPLOT - standard.c */
/*[
* Copyright 1986 - 1993, 1998, 2004 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 complete modified source code. Modifications are to
* be distributed as patches to the released version. Permission to
* distribute binaries produced by compiling modified sources is granted,
* provided you
* 1. distribute the corresponding source modifications from the
* released version in the form of a patch file along with the binaries,
* 2. add special version identification to distinguish your version
* in addition to the base release version number,
* 3. provide your name and address as the primary contact for the
* support of your modified version, and
* 4. retain our contact information in regard to use of the base
* software.
* Permission to distribute the released version of the source code along
* with corresponding source modifications in the form of a patch file is
* granted with same provisions 2 through 4 for binary distributions.
*
* This software is provided "as is" without express or implied warranty
* to the extent permitted by applicable law.
]*/
#include "standard.h"
#include "gadgets.h" /* for 'ang2rad' and 'zero' */
#include "gp_time.h" /* needed by f_tmsec() and friendsa */
#include "util.h" /* for int_error() */
static double carlson_elliptic_rc(double x,double y);
static double carlson_elliptic_rf(double x,double y,double z);
static double carlson_elliptic_rd(double x,double y,double z);
static double carlson_elliptic_rj(double x,double y,double z,double p);
static double jzero __PROTO((double x));
static double pzero __PROTO((double x));
static double qzero __PROTO((double x));
static double yzero __PROTO((double x));
static double rj0 __PROTO((double x));
static double ry0 __PROTO((double x));
static double jone __PROTO((double x));
static double pone __PROTO((double x));
static double qone __PROTO((double x));
static double yone __PROTO((double x));
static double rj1 __PROTO((double x));
static double ry1 __PROTO((double x));
/* 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 GPFAR 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 GPFAR 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 GPFAR ppzero[] = {
0.2277909019730468430227002627e+5,
0.4134538663958076579678016384e+5,
0.2117052338086494432193395727e+5,
0.348064864432492703474453111e+4,
0.15376201909008354295771715e+3,
0.889615484242104552360748e+0
};
static double GPFAR 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 GPFAR pqzero[] = {
-0.8922660020080009409846916e+2,
-0.18591953644342993800252169e+3,
-0.11183429920482737611262123e+3,
-0.2230026166621419847169915e+2,
-0.124410267458356384591379e+1,
-0.8803330304868075181663e-2,
};
static double GPFAR 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 GPFAR 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 GPFAR 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 GPFAR 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 GPFAR 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 GPFAR ppone[] = {
0.352246649133679798341724373e+5,
0.62758845247161281269005675e+5,
0.313539631109159574238669888e+5,
0.49854832060594338434500455e+4,
0.2111529182853962382105718e+3,
0.12571716929145341558495e+1
};
static double GPFAR 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 GPFAR pqone[] = {
0.3511751914303552822533318e+3,
0.7210391804904475039280863e+3,
0.4259873011654442389886993e+3,
0.831898957673850827325226e+2,
0.45681716295512267064405e+1,
0.3532840052740123642735e-1
};
static double GPFAR 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 GPFAR 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 GPFAR 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
};
/*
* Make all the following internal routines perform autoconversion
* from string to numeric value.
*/
#define pop(x) pop_or_convert_from_string(x)
void
f_real(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
push(Gcomplex(&a, real(pop(&a)), 0.0));
}
void
f_imag(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
push(Gcomplex(&a, imag(pop(&a)), 0.0));
}
/* ang2rad is 1 if we are in radians, or pi/180 if we are in degrees */
void
f_arg(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
push(Gcomplex(&a, angle(pop(&a)) / ang2rad, 0.0));
}
void
f_conjg(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
push(Gcomplex(&a, real(&a), -imag(&a)));
}
/* ang2rad is 1 if we are in radians, or pi/180 if we are in degrees */
void
f_sin(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
push(Gcomplex(&a, sin(ang2rad * real(&a)) * cosh(ang2rad * imag(&a)), cos(ang2rad * real(&a)) * sinh(ang2rad * imag(&a))));
}
void
f_cos(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
push(Gcomplex(&a, cos(ang2rad * real(&a)) * cosh(ang2rad * imag(&a)), -sin(ang2rad * real(&a)) * sinh(ang2rad * imag(&a))));
}
void
f_tan(union argument *arg)
{
struct value a;
double den;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (imag(&a) == 0.0)
push(Gcomplex(&a, tan(ang2rad * real(&a)), 0.0));
else {
den = cos(2 * ang2rad * real(&a)) + cosh(2 * ang2rad * imag(&a));
if (den == 0.0) {
undefined = TRUE;
push(&a);
} else
push(Gcomplex(&a, sin(2 * ang2rad * real(&a)) / den, sinh(2 * ang2rad * imag(&a)) / den));
}
}
void
f_asin(union argument *arg)
{
struct value a;
double alpha, beta, x, y;
int ysign;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = real(&a);
y = imag(&a);
if (y == 0.0 && fabs(x) <= 1.0) {
push(Gcomplex(&a, asin(x) / ang2rad, 0.0));
} else if (x == 0.0) {
push(Gcomplex(&a, 0.0, -log(-y + sqrt(y * y + 1)) / ang2rad));
} else {
beta = sqrt((x + 1) * (x + 1) + y * y) / 2 - sqrt((x - 1) * (x - 1) + y * y) / 2;
if (beta > 1)
beta = 1; /* Avoid rounding error problems */
alpha = sqrt((x + 1) * (x + 1) + y * y) / 2 + sqrt((x - 1) * (x - 1) + y * y) / 2;
ysign = (y >= 0) ? 1 : -1;
push(Gcomplex(&a, asin(beta) / ang2rad, ysign * log(alpha + sqrt(alpha * alpha - 1)) / ang2rad));
}
}
void
f_acos(union argument *arg)
{
struct value a;
double x, y;
double ysign;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = real(&a);
y = imag(&a);
if (y == 0.0 && fabs(x) <= 1.0) {
/* real result */
push(Gcomplex(&a, acos(x) / ang2rad, 0.0));
} else {
double alpha = sqrt((x + 1) * (x + 1) + y * y) / 2
+ sqrt((x - 1) * (x - 1) + y * y) / 2;
double beta = sqrt((x + 1) * (x + 1) + y * y) / 2
- sqrt((x - 1) * (x - 1) + y * y) / 2;
if (beta > 1)
beta = 1; /* Avoid rounding error problems */
else if (beta < -1)
beta = -1;
ysign = (y >= 0) ? 1 : -1;
push(Gcomplex(&a, acos(beta) / ang2rad,
-ysign * log(alpha + sqrt(alpha * alpha - 1)) / ang2rad));
}
}
void
f_atan(union argument *arg)
{
struct value a;
double x, y, u, v, w, z;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = real(&a);
y = imag(&a);
if (y == 0.0)
push(Gcomplex(&a, atan(x) / ang2rad, 0.0));
else if (x == 0.0 && fabs(y) >= 1.0) {
undefined = TRUE;
push(Gcomplex(&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(Gcomplex(&a, 0.5 * z / ang2rad, w));
}
}
/* real parts only */
void
f_atan2(union argument *arg)
{
struct value a;
double x;
double y;
(void) arg; /* avoid -Wunused warning */
x = real(pop(&a));
y = real(pop(&a));
if (x == 0.0 && y == 0.0) {
undefined = TRUE;
push(Ginteger(&a, 0));
}
push(Gcomplex(&a, atan2(y, x) / ang2rad, 0.0));
}
void
f_sinh(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
push(Gcomplex(&a, sinh(real(&a)) * cos(imag(&a)), cosh(real(&a)) * sin(imag(&a))));
}
void
f_cosh(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
push(Gcomplex(&a, cosh(real(&a)) * cos(imag(&a)), sinh(real(&a)) * sin(imag(&a))));
}
void
f_tanh(union argument *arg)
{
struct value a;
double den;
double real_2arg, imag_2arg;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
real_2arg = 2. * real(&a);
imag_2arg = 2. * imag(&a);
#ifdef E_MINEXP
if (-fabs(real_2arg) < E_MINEXP) {
push(Gcomplex(&a, real_2arg < 0 ? -1.0 : 1.0, 0.0));
return;
}
#else
{
int old_errno = errno;
if (exp(-fabs(real_2arg)) == 0.0) {
/* some libm's will raise a silly ERANGE in cosh() and sin() */
errno = old_errno;
push(Gcomplex(&a, real_2arg < 0 ? -1.0 : 1.0, 0.0));
return;
}
}
#endif
den = cosh(real_2arg) + cos(imag_2arg);
push(Gcomplex(&a, sinh(real_2arg) / den, sin(imag_2arg) / den));
}
void
f_asinh(union argument *arg)
{
struct value a; /* asinh(z) = -I*asin(I*z) */
double alpha, beta, x, y;
int ysign;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = -imag(&a);
y = real(&a);
if (y == 0.0 && fabs(x) <= 1.0) {
push(Gcomplex(&a, 0.0, -asin(x) / ang2rad));
} else if (y == 0.0) {
push(Gcomplex(&a, 0.0, 0.0));
undefined = TRUE;
} else if (x == 0.0) {
push(Gcomplex(&a, log(y + sqrt(y * y + 1)) / ang2rad, 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;
ysign = (y >= 0) ? 1 : -1;
push(Gcomplex(&a, ysign * log(alpha + sqrt(alpha * alpha - 1)) / ang2rad, -asin(beta) / ang2rad));
}
}
void
f_acosh(union argument *arg)
{
struct value a;
double alpha, beta, x, y; /* acosh(z) = I*acos(z) */
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = real(&a);
y = imag(&a);
if (y == 0.0 && fabs(x) <= 1.0) {
push(Gcomplex(&a, 0.0, acos(x) / ang2rad));
} else if (y == 0) {
push(Gcomplex(&a, log(x + sqrt(x * x - 1)) / ang2rad, 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(Gcomplex(&a, log(alpha + sqrt(alpha * alpha - 1)) / ang2rad,
(y<0 ? -1 : 1) * acos(beta) / ang2rad));
}
}
void
f_atanh(union argument *arg)
{
struct value a;
double x, y, u, v, w, z; /* atanh(z) = -I*atan(I*z) */
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
x = -imag(&a);
y = real(&a);
if (y == 0.0)
push(Gcomplex(&a, 0.0, -atan(x) / ang2rad));
else if (x == 0.0 && fabs(y) >= 1.0) {
undefined = TRUE;
push(Gcomplex(&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(Gcomplex(&a, w, -0.5 * z / ang2rad));
}
}
void
f_ellip_first(union argument *arg)
{
struct value a;
double ak,q;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do elliptic integrals of reals");
ak=real(&a);
q=(1.0-ak)*(1.0+ak);
if (q > 0.0)
push(Gcomplex(&a,carlson_elliptic_rf(0.0,q,1.0) ,0.0));
else {
push(&a);
undefined=TRUE;
}
}
void
f_ellip_second(union argument *arg)
{
struct value a;
double ak,q,e;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do elliptic integrals of reals");
ak=real(&a);
q=(1.0-ak)*(1.0+ak);
if (q > 0.0) {
e=carlson_elliptic_rf(0.0,q,1.0)-(ak*ak)*carlson_elliptic_rd(0.0,q,1.0)/3.0;
push(Gcomplex(&a,e,0.0));
} else if (q < 0.0) {
undefined=TRUE;
push(&a);
} else {
e=1.0;
push(Gcomplex(&a,e,0.0));
}
}
void
f_ellip_third(union argument *arg)
{
struct value a1,a2;
double ak,en,q;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a1);
(void) pop(&a2);
if (fabs(imag(&a1)) > zero || fabs(imag(&a2)) > zero)
int_error(NO_CARET, "can only do elliptic integrals of reals");
ak=real(&a1);
en=real(&a2);
q=(1.0-ak)*(1.0+ak);
if (q > 0.0 && en < 1.0)
push(Gcomplex(&a2, carlson_elliptic_rf(0.0,q,1.0)+en*carlson_elliptic_rj(0.0,q,1.0,1.0-en)/3.0, 0.0));
else {
undefined=TRUE;
push(&a1);
}
}
void
f_int(union argument *arg)
{
struct value a;
double foo = real(pop(&a));
(void) arg; /* avoid -Wunused warning */
if (a.type == NOTDEFINED || isnan(foo)) {
push(Gcomplex(&a, not_a_number(), 0.0));
undefined = TRUE;
} else
push(Ginteger(&a, (int)foo));
}
#define BAD_DEFAULT default: int_error(NO_CARET, "internal error : argument neither INT or CMPLX")
void
f_abs(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
switch (a.type) {
case INTGR:
push(Ginteger(&a, abs(a.v.int_val)));
break;
case CMPLX:
push(Gcomplex(&a, magnitude(&a), 0.0));
break;
BAD_DEFAULT;
}
}
void
f_sgn(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
switch (a.type) {
case INTGR:
push(Ginteger(&a, (a.v.int_val > 0) ? 1 :
(a.v.int_val < 0) ? -1 : 0));
break;
case CMPLX:
push(Ginteger(&a, (a.v.cmplx_val.real > 0.0) ? 1 :
(a.v.cmplx_val.real < 0.0) ? -1 : 0));
break;
BAD_DEFAULT;
}
}
void
f_sqrt(union argument *arg)
{
struct value a;
double mag;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
mag = sqrt(magnitude(&a));
if (imag(&a) == 0.0) {
if (real(&a) < 0.0)
push(Gcomplex(&a, 0.0, mag));
else
push(Gcomplex(&a, mag, 0.0));
} else {
/* -pi < ang < pi, so real(sqrt(z)) >= 0 */
double ang = angle(&a) / 2.0;
push(Gcomplex(&a, mag * cos(ang), mag * sin(ang)));
}
}
void
f_exp(union argument *arg)
{
struct value a;
double mag, ang;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
mag = gp_exp(real(&a));
ang = imag(&a);
push(Gcomplex(&a, mag * cos(ang), mag * sin(ang)));
}
void
f_log10(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (magnitude(&a) == 0.0) {
undefined = TRUE;
push(&a);
} else
push(Gcomplex(&a, log(magnitude(&a)) / M_LN10, angle(&a) / M_LN10));
}
void
f_log(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (magnitude(&a) == 0.0) {
undefined = TRUE;
push(&a);
} else
push(Gcomplex(&a, log(magnitude(&a)), angle(&a)));
}
void
f_floor(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
switch (a.type) {
case INTGR:
push(Ginteger(&a, (int) floor((double) a.v.int_val)));
break;
case CMPLX:
push(Ginteger(&a, (int) floor(a.v.cmplx_val.real)));
break;
BAD_DEFAULT;
}
}
void
f_ceil(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
switch (a.type) {
case INTGR:
push(Ginteger(&a, (int) ceil((double) a.v.int_val)));
break;
case CMPLX:
push(Ginteger(&a, (int) ceil(a.v.cmplx_val.real)));
break;
BAD_DEFAULT;
}
}
/* Terminate the autoconversion from string to numeric values */
#undef pop
/* EAM - replacement for defined(foo) + f_pushv + f_isvar
* implements exists("foo") instead
*/
void
f_exists(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (a.type == STRING) {
struct udvt_entry *udv = add_udv_by_name(a.v.string_val);
gpfree_string(&a);
push(Ginteger(&a, udv->udv_value.type == NOTDEFINED ? 0 : 1));
} else {
push(Ginteger(&a, 0));
}
}
/* bessel function approximations */
static double
jzero(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);
}
static double
pzero(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);
}
static double
qzero(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);
}
static double
yzero(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);
}
static double
rj0(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)));
}
static double
ry0(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)));
}
static double
jone(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);
}
static double
pone(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);
}
static double
qone(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);
}
static double
yone(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);
}
static double
rj1(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);
}
}
static double
ry1(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)));
}
/* FIXME HBB 20010726: should bessel functions really call int_error,
* right in the middle of evaluating some mathematical expression?
* Couldn't they just flag 'undefined', or ignore the real part of the
* complex number? */
void
f_besj0(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do bessel functions of reals");
push(Gcomplex(&a, rj0(real(&a)), 0.0));
}
void
f_besj1(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do bessel functions of reals");
push(Gcomplex(&a, rj1(real(&a)), 0.0));
}
void
f_besy0(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do bessel functions of reals");
if (real(&a) > 0.0)
push(Gcomplex(&a, ry0(real(&a)), 0.0));
else {
push(Gcomplex(&a, 0.0, 0.0));
undefined = TRUE;
}
}
void
f_besy1(union argument *arg)
{
struct value a;
(void) arg; /* avoid -Wunused warning */
(void) pop(&a);
if (fabs(imag(&a)) > zero)
int_error(NO_CARET, "can only do bessel functions of reals");
if (real(&a) > 0.0)
push(Gcomplex(&a, ry1(real(&a)), 0.0));
else {
push(Gcomplex(&a, 0.0, 0.0));
undefined = TRUE;
}
}
/* functions for accessing fields from tm structure, for time series
* they are all the same, so define a macro
*/
#define TIMEFUNC(name, field) \
void \
name(union argument *arg) \
{ \
struct value a; \
struct tm tm; \
\
(void) arg; /* avoid -Wunused warning */ \
(void) pop(&a); \
ggmtime(&tm, real(&a)); \
push(Gcomplex(&a, (double)tm.field, 0.0)); \
}
TIMEFUNC( f_tmsec, tm_sec)
TIMEFUNC( f_tmmin, tm_min)
TIMEFUNC( f_tmhour, tm_hour)
TIMEFUNC( f_tmmday, tm_mday)
TIMEFUNC( f_tmmon, tm_mon)
TIMEFUNC( f_tmyear, tm_year)
TIMEFUNC( f_tmwday, tm_wday)
TIMEFUNC( f_tmyday, tm_yday)
/*****************************************************************************/
#define SQR(a) ((a)*(a))
#define C1 0.3
#define C2 (1.0/7.0)
#define C3 0.375
#define C4 (9.0/22.0)
static double
carlson_elliptic_rc(double x,double y)
{
double alamb,ave,s,w,xt,yt,ans;
if (y > 0.0) {
xt=x;
yt=y;
w=1.0;
} else {
xt=x-y;
yt = -y;
w=sqrt(x)/sqrt(xt);
}
do {
alamb=2.0*sqrt(xt)*sqrt(yt)+yt;
xt=0.25*(xt+alamb);
yt=0.25*(yt+alamb);
ave=(1.0/3.0)*(xt+yt+yt);
s=(yt-ave)/ave;
} while (fabs(s) > 0.0012);
ans=w*(1.0+s*s*(C1+s*(C2+s*(C3+s*C4))))/sqrt(ave);
return(ans);
}
#undef C4
#undef C3
#undef C2
#undef C1
static double
carlson_elliptic_rf(double x,double y,double z)
{
double alamb,ave,delx,dely,delz,e2,e3,sqrtx,sqrty,sqrtz,xt,yt,zt;
xt=x;
yt=y;
zt=z;
do {
sqrtx=sqrt(xt);
sqrty=sqrt(yt);
sqrtz=sqrt(zt);
alamb=sqrtx*(sqrty+sqrtz)+sqrty*sqrtz;
xt=0.25*(xt+alamb);
yt=0.25*(yt+alamb);
zt=0.25*(zt+alamb);
ave=(1.0/3.0)*(xt+yt+zt);
delx=(ave-xt)/ave;
dely=(ave-yt)/ave;
delz=(ave-zt)/ave;
} while (fabs(delx) > 0.0025 || fabs(dely) > 0.0025 || fabs(delz) > 0.0025);
e2=delx*dely-delz*delz;
e3=delx*dely*delz;
return((1.0+((1.0/24.0)*e2-(0.1)-(3.0/44.0)*e3)*e2+(1.0/14.0)*e3)/sqrt(ave));
}
#define C1 (3.0/14.0)
#define C2 (1.0/6.0)
#define C3 (9.0/22.0)
#define C4 (3.0/26.0)
#define C5 (0.25*C3)
#define C6 (1.5*C4)
static double
carlson_elliptic_rd(double x,double y,double z)
{
double alamb,ave,delx,dely,delz,ea,eb,ec,ed,ee,fac,
sqrtx,sqrty,sqrtz,sum,xt,yt,zt,ans;
xt=x;
yt=y;
zt=z;
sum=0.0;
fac=1.0;
do {
sqrtx=sqrt(xt);
sqrty=sqrt(yt);
sqrtz=sqrt(zt);
alamb=sqrtx*(sqrty+sqrtz)+sqrty*sqrtz;
sum+=fac/(sqrtz*(zt+alamb));
fac=0.25*fac;
xt=0.25*(xt+alamb);
yt=0.25*(yt+alamb);
zt=0.25*(zt+alamb);
ave=0.2*(xt+yt+3.0*zt);
delx=(ave-xt)/ave;
dely=(ave-yt)/ave;
delz=(ave-zt)/ave;
} while (fabs(delx) > 0.0015 || fabs(dely) > 0.0015 || fabs(delz) > 0.0015);
ea=delx*dely;
eb=delz*delz;
ec=ea-eb;
ed=ea-6.0*eb;
ee=ed+ec+ec;
ans=3.0*sum+fac*(1.0+ed*(-C1+C5*ed-C6*delz*ee)
+delz*(C2*ee+delz*(-C3*ec+delz*C4*ea)))/(ave*sqrt(ave));
return(ans);
}
#undef C6
#undef C5
#undef C4
#undef C3
#undef C2
#undef C1
#define C1 (3.0/14.0)
#define C2 (1.0/3.0)
#define C3 (3.0/22.0)
#define C4 (3.0/26.0)
#define C5 (0.75*C3)
#define C6 (1.5*C4)
#define C7 (0.5*C2)
#define C8 (C3+C3)
static double
carlson_elliptic_rj(double x,double y,double z,double p)
{
double a,alamb,alpha,ans,ave,b,beta,delp,delx,dely,delz,ea,eb,ec,
ed,ee,fac,pt,rcx,rho,sqrtx,sqrty,sqrtz,sum,tau,xt,yt,zt;
sum=0.0;
fac=1.0;
if (p > 0.0) {
xt=x;
yt=y;
zt=z;
pt=p;
a=b=rcx=0.0;
} else {
xt=GPMIN(GPMIN(x,y),z);
zt=GPMAX(GPMAX(x,y),z);
yt=x+y+z-xt-zt;
a=1.0/(yt-p);
b=a*(zt-yt)*(yt-xt);
pt=yt+b;
rho=xt*zt/yt;
tau=p*pt/yt;
rcx=carlson_elliptic_rc(rho,tau);
}
do {
sqrtx=sqrt(xt);
sqrty=sqrt(yt);
sqrtz=sqrt(zt);
alamb=sqrtx*(sqrty+sqrtz)+sqrty*sqrtz;
alpha=SQR(pt*(sqrtx+sqrty+sqrtz)+sqrtx*sqrty*sqrtz);
beta=pt*SQR(pt+alamb);
sum += fac*carlson_elliptic_rc(alpha,beta);
fac=0.25*fac;
xt=0.25*(xt+alamb);
yt=0.25*(yt+alamb);
zt=0.25*(zt+alamb);
pt=0.25*(pt+alamb);
ave=0.2*(xt+yt+zt+pt+pt);
delx=(ave-xt)/ave;
dely=(ave-yt)/ave;
delz=(ave-zt)/ave;
delp=(ave-pt)/ave;
} while (fabs(delx)>0.0015 || fabs(dely)>0.0015 || fabs(delz)>0.0015 || fabs(delp)>0.0015);
ea=delx*(dely+delz)+dely*delz;
eb=delx*dely*delz;
ec=delp*delp;
ed=ea-3.0*ec;
ee=eb+2.0*delp*(ea-ec);
ans=3.0*sum+fac*(1.0+ed*(-C1+C5*ed-C6*ee)+eb*(C7+delp*(-C8+delp*C4))
+delp*ea*(C2-delp*C3)-C2*delp*ec)/(ave*sqrt(ave));
if (p <= 0.0)
ans=a*(b*ans+3.0*(rcx-carlson_elliptic_rf(xt,yt,zt)));
return(ans);
}
#undef C6
#undef C5
#undef C4
#undef C3
#undef C2
#undef C1
#undef C8
#undef C7
#undef SQR
/*****************************************************************************/