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MPMATH_C.C
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1995-03-08
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/* MPMath_c.c (C) 1989, Mark C. Peterson, CompuServe [70441,3353]
All rights reserved.
Code may be used in any program provided the author is credited
either during program execution or in the documentation. Source
code may be distributed only in combination with public domain or
shareware source code. Source code may be modified provided the
copyright notice and this message is left unchanged and all
modifications are clearly documented.
I would appreciate a copy of any work which incorporates this code,
however this is optional.
Mark C. Peterson
405-C Queen St. Suite #181
Southington, CT 06489
(203) 276-9721
*/
#include <stdlib.h>
/* This now in prototyp.h */
/*
#include "mpmath.h"
*/
#include "prototyp.h"
#ifndef XFRACT
struct MP *MPsub(struct MP x, struct MP y) {
y.Exp ^= 0x8000;
return(MPadd(x, y));
}
/* added by TW */
struct MP *MPsub086(struct MP x, struct MP y) {
y.Exp ^= 0x8000;
return(MPadd086(x, y));
}
/* added by TW */
struct MP *MPsub386(struct MP x, struct MP y) {
y.Exp ^= 0x8000;
return(MPadd386(x, y));
}
struct MP *MPabs(struct MP x) {
Ans = x;
Ans.Exp &= 0x7fff;
return(&Ans);
}
struct MPC MPCsqr(struct MPC x) {
struct MPC z;
z.x = *pMPsub(*pMPmul(x.x, x.x), *pMPmul(x.y, x.y));
z.y = *pMPmul(x.x, x.y);
z.y.Exp++;
return(z);
}
struct MP MPCmod(struct MPC x) {
return(*pMPadd(*pMPmul(x.x, x.x), *pMPmul(x.y, x.y)));
}
struct MPC MPCmul(struct MPC x, struct MPC y) {
struct MPC z;
z.x = *pMPsub(*pMPmul(x.x, y.x), *pMPmul(x.y, y.y));
z.y = *pMPadd(*pMPmul(x.x, y.y), *pMPmul(x.y, y.x));
return(z);
}
struct MPC MPCdiv(struct MPC x, struct MPC y) {
struct MP mod;
mod = MPCmod(y);
y.y.Exp ^= 0x8000;
y.x = *pMPdiv(y.x, mod);
y.y = *pMPdiv(y.y, mod);
return(MPCmul(x, y));
}
struct MPC MPCadd(struct MPC x, struct MPC y) {
struct MPC z;
z.x = *pMPadd(x.x, y.x);
z.y = *pMPadd(x.y, y.y);
return(z);
}
struct MPC MPCsub(struct MPC x, struct MPC y) {
struct MPC z;
z.x = *pMPsub(x.x, y.x);
z.y = *pMPsub(x.y, y.y);
return(z);
}
struct MPC MPCone = { 0x3fff, 0x80000000l, 0, 0l };
struct MPC MPCpow(struct MPC x, int exp) {
struct MPC z;
struct MPC zz;
if(exp & 1)
z = x;
else
z = MPCone;
exp >>= 1;
while(exp) {
zz.x = *pMPsub(*pMPmul(x.x, x.x), *pMPmul(x.y, x.y));
zz.y = *pMPmul(x.x, x.y);
zz.y.Exp++;
x = zz;
if(exp & 1) {
zz.x = *pMPsub(*pMPmul(z.x, x.x), *pMPmul(z.y, x.y));
zz.y = *pMPadd(*pMPmul(z.x, x.y), *pMPmul(z.y, x.x));
z = zz;
}
exp >>= 1;
}
return(z);
}
int MPCcmp(struct MPC x, struct MPC y) {
struct MPC z;
if(pMPcmp(x.x, y.x) || pMPcmp(x.y, y.y)) {
z.x = MPCmod(x);
z.y = MPCmod(y);
return(pMPcmp(z.x, z.y));
}
else
return(0);
}
_CMPLX MPC2cmplx(struct MPC x) {
_CMPLX z;
z.x = *pMP2d(x.x);
z.y = *pMP2d(x.y);
return(z);
}
struct MPC cmplx2MPC(_CMPLX z) {
struct MPC x;
x.x = *pd2MP(z.x);
x.y = *pd2MP(z.y);
return(x);
}
/* function pointer versions added by Tim Wegner 12/07/89 */
/* int (*ppMPcmp)() = MPcmp086; */
int (*pMPcmp)(struct MP x, struct MP y) = MPcmp086;
struct MP *(*pMPmul)(struct MP x, struct MP y)= MPmul086;
struct MP *(*pMPdiv)(struct MP x, struct MP y)= MPdiv086;
struct MP *(*pMPadd)(struct MP x, struct MP y)= MPadd086;
struct MP *(*pMPsub)(struct MP x, struct MP y)= MPsub086;
struct MP *(*pd2MP)(double x) = d2MP086 ;
double *(*pMP2d)(struct MP m) = MP2d086 ;
/* struct MP *(*pfg2MP)(long x, int fg) = fg2MP086; */
void setMPfunctions(void) {
if(cpu == 386)
{
pMPmul = MPmul386;
pMPdiv = MPdiv386;
pMPadd = MPadd386;
pMPsub = MPsub386;
pMPcmp = MPcmp386;
pd2MP = d2MP386 ;
pMP2d = MP2d386 ;
/* pfg2MP = fg2MP386; */
}
else
{
pMPmul = MPmul086;
pMPdiv = MPdiv086;
pMPadd = MPadd086;
pMPsub = MPsub086;
pMPcmp = MPcmp086;
pd2MP = d2MP086 ;
pMP2d = MP2d086 ;
/* pfg2MP = fg2MP086; */
}
}
#endif /* XFRACT */
#ifndef sqr
#define sqr(x) ((x)*(x))
#endif
_CMPLX ComplexPower(_CMPLX xx, _CMPLX yy) {
_CMPLX z, cLog, t;
double e2x, siny, cosy;
/* fixes power bug - if any complaints, backwards compatibility hook
goes here TIW 3/95 */
if(debugflag != 94)
if(xx.x == 0 && xx.y == 0) {
z.x = z.y = 0.0;
return(z);
}
FPUcplxlog(&xx, &cLog);
FPUcplxmul(&cLog, &yy, &t);
if(fpu == 387)
FPUcplxexp387(&t, &z);
else {
if(t.x < -690)
e2x = 0;
else
e2x = exp(t.x);
FPUsincos(&t.y, &siny, &cosy);
z.x = e2x * cosy;
z.y = e2x * siny;
}
return(z);
}
/*
The following Complex function routines added by Tim Wegner November 1994.
*/
#define Sqrtz(z,rz) (*(rz) = ComplexSqrtFloat((z).x, (z).y))
/* rz=Arcsin(z)=-i*Log{i*z+sqrt(1-z*z)} */
void Arcsinz(_CMPLX z,_CMPLX *rz)
{
_CMPLX tempz1,tempz2;
if(z.y == 0 && z.x > 1)
{
rz->x = HUGE_VAL;
rz->y = 0;
return;
}
FPUcplxmul( &z, &z, &tempz1);
tempz1.x = 1 - tempz1.x; tempz1.y = -tempz1.y; /* tempz1 = 1 - tempz1 */
Sqrtz( tempz1, &tempz1);
tempz2.x = -z.y; tempz2.y = z.x; /* tempz2 = i*z */
tempz1.x += tempz2.x; tempz1.y += tempz2.y; /* tempz1 += tempz2 */
FPUcplxlog( &tempz1, &tempz1);
rz->x = tempz1.y; rz->y = -tempz1.x; /* rz = (-i)*tempz1 */
} /* end. Arcsinz */
/* rz=Arccos(z)=-i*Log{z+sqrt(z*z-1)} */
void Arccosz(_CMPLX z,_CMPLX *rz)
{
_CMPLX temp;
FPUcplxmul( &z, &z, &temp);
temp.x -= 1; /* temp = temp - 1 */
Sqrtz( temp, &temp);
temp.x += z.x; temp.y += z.y; /* temp = z + temp */
FPUcplxlog( &temp, &temp);
rz->x = temp.y; rz->y = -temp.x; /* rz = (-i)*tempz1 */
} /* end. Arccosz */
void Arcsinhz(_CMPLX z,_CMPLX *rz)
{
_CMPLX temp;
FPUcplxmul( &z, &z, &temp);
temp.x += 1; /* temp = temp + 1 */
Sqrtz( temp, &temp);
temp.x += z.x; temp.y += z.y; /* temp = z + temp */
FPUcplxlog( &temp, rz);
} /* end. Arcsinhz */
/* rz=Arccosh(z)=Log(z+sqrt(z*z-1)} */
void Arccoshz(_CMPLX z,_CMPLX *rz)
{
_CMPLX tempz;
FPUcplxmul( &z, &z, &tempz);
tempz.x -= 1; /* tempz = tempz - 1 */
Sqrtz( tempz, &tempz);
tempz.x = z.x + tempz.x; tempz.y = z.y + tempz.y; /* tempz = z + tempz */
FPUcplxlog( &tempz, rz);
} /* end. Arccoshz */
/* rz=Arctanh(z)=1/2*Log{(1+z)/(1-z)} */
void Arctanhz(_CMPLX z,_CMPLX *rz)
{
_CMPLX temp0,temp1,temp2;
if( z.x == 0.0){
rz->x = 0;
rz->y = atan( z.y);
return;
}
else{
if( fabs(z.x) == 1.0 && z.y == 0.0){
return;
}
else if( fabs( z.x) < 1.0 && z.y == 0.0){
rz->x = log((1+z.x)/(1-z.x))/2;
rz->y = 0;
return;
}
else{
temp0.x = 1 + z.x; temp0.y = z.y; /* temp0 = 1 + z */
temp1.x = 1 - z.x; temp1.y = -z.y; /* temp1 = 1 - z */
FPUcplxdiv( &temp0, &temp1, &temp2);
FPUcplxlog( &temp2, &temp2);
rz->x = .5*temp2.x; rz->y = .5*temp2.y; /* rz = .5*temp2 */
return;
}
}
} /* end. Arctanhz */
/* rz=Arctan(z)=i/2*Log{(1-i*z)/(1+i*z)} */
void Arctanz(_CMPLX z,_CMPLX *rz)
{
_CMPLX temp0,temp1,temp2,temp3;
if( z.x == 0.0 && z.y == 0.0)
rz->x = rz->y = 0;
else if( z.x != 0.0 && z.y == 0.0){
rz->x = atan( z.x);
rz->y = 0;
}
else if( z.x == 0.0 && z.y != 0.0){
temp0.x = z.y; temp0.y = 0.0;
Arctanhz( temp0, &temp0);
rz->x = -temp0.y; rz->y = temp0.x; /* i*temp0 */
}
else if( z.x != 0.0 && z.y != 0.0){
temp0.x = -z.y; temp0.y = z.x; /* i*z */
temp1.x = 1 - temp0.x; temp1.y = -temp0.y; /* temp1 = 1 - temp0 */
temp2.x = 1 + temp0.x; temp2.y = temp0.y; /* temp2 = 1 + temp0 */
FPUcplxdiv( &temp1, &temp2, &temp3);
FPUcplxlog( &temp3, &temp3);
rz->x = -temp3.y*.5; rz->y = .5*temp3.x; /* .5*i*temp0 */
}
} /* end. Arctanz */
#define SinCosFudge 0x10000L
#ifdef LONGSQRT
long lsqrt(long f)
{
int N;
unsigned long y0, z;
static long a=0, b=0, c=0; /* constant factors */
if (f == 0)
return f;
if (f < 0)
return 0;
if (a==0) /* one-time compute consts */
{
a = (long)(fudge * .41731);
b = (long)(fudge * .59016);
c = (long)(fudge * .7071067811);
}
N = 0;
while (f & 0xff000000L) /* shift arg f into the */
{ /* range: 0.5 <= f < 1 */
N++;
f /= 2;
}
while (!(f & 0xff800000L))
{
N--;
f *= 2;
}
y0 = a + multiply(b, f, bitshift); /* Newton's approximation */
z = y0 + divide (f, y0, bitshift);
y0 = (z>>2) + divide(f, z, bitshift);
if (N % 2)
{
N++;
y0 = multiply(c,y0, bitshift);
}
N /= 2;
if (N >= 0)
return y0 << N; /* correct for shift above */
else
return y0 >> -N;
}
#endif
LCMPLX ComplexSqrtLong(long x, long y)
{
double mag, theta;
long maglong, thetalong;
LCMPLX result;
#ifndef LONGSQRT
mag = sqrt(sqrt(((double) multiply(x,x,bitshift))/fudge +
((double) multiply(y,y,bitshift))/ fudge));
maglong = (long)(mag * fudge);
#else
maglong = lsqrt(lsqrt(multiply(x,x,bitshift)+multiply(y,y,bitshift)));
#endif
theta = atan2((double) y/fudge, (double) x/fudge)/2;
thetalong = (long)(theta * SinCosFudge);
SinCos086(thetalong, &result.y, &result.x);
result.x = multiply(result.x << (bitshift - 16), maglong, bitshift);
result.y = multiply(result.y << (bitshift - 16), maglong, bitshift);
return result;
}
_CMPLX ComplexSqrtFloat(double x, double y)
{
double mag;
double theta;
_CMPLX result;
if(x == 0.0 && y == 0.0)
result.x = result.y = 0.0;
else
{
mag = sqrt(sqrt(x*x + y*y));
theta = atan2(y, x) / 2;
FPUsincos(&theta, &result.y, &result.x);
result.x *= mag;
result.y *= mag;
}
return result;
}
/***** FRACTINT specific routines and variables *****/
#ifndef TESTING_MATH
BYTE far *LogTable = (BYTE far *)0;
int MaxLTSize;
/* int LogFlag;
LogFlag == 1 -- standard log palettes
LogFlag == -1 -- 'old' log palettes
LogFlag > 1 -- compress counts < LogFlag into color #1
LogFlag < -1 -- use quadratic palettes based on square roots && compress
*/
void SetupLogTable(void) {
float l, f, c, m;
unsigned n, prev, limit, lf;
if (LogFlag > -2) {
lf = (LogFlag > 1) ? LogFlag : 0;
if (lf >= (unsigned int)MaxLTSize)
lf = MaxLTSize - 1;
Fg2Float((long)(MaxLTSize-lf), 0, m);
fLog14(m, m);
Fg2Float((long)(colors-(lf?2:1)), 0, c);
fDiv(m, c, m);
for (prev = 1; prev <= lf; prev++)
LogTable[prev] = 1;
for (n = (lf?2:1); n < (unsigned int)colors; n++) {
Fg2Float((long)n, 0, f);
fMul16(f, m, f);
fExp14(f, l);
limit = (unsigned int)Float2Fg(l, 0) + lf;
if (limit > (unsigned int)MaxLTSize || n == (unsigned int)(colors-1))
limit = MaxLTSize;
while (prev <= limit)
LogTable[prev++] = (BYTE)n;
}
} else {
if ((lf = 0 - LogFlag) >= (unsigned int)MaxLTSize)
lf = MaxLTSize - 1;
Fg2Float((long)(MaxLTSize-lf), 0, m);
fSqrt14(m, m);
Fg2Float((long)(colors-2), 0, c);
fDiv(m, c, m);
for (prev = 1; prev <= lf; prev++)
LogTable[prev] = 1;
for (n = 2; n < (unsigned int)colors; n++) {
Fg2Float((long)n, 0, f);
fMul16(f, m, f);
fMul16(f, f, l);
limit = (unsigned int)(Float2Fg(l, 0) + lf);
if (limit > (unsigned int)MaxLTSize || n == (unsigned int)(colors-1))
limit = MaxLTSize;
while (prev <= limit)
LogTable[prev++] = (BYTE)n;
}
}
LogTable[0] = 0;
if (LogFlag != -1)
for (n = 1; n < (unsigned int)MaxLTSize; n++) /* spread top to incl unused colors */
if (LogTable[n] > LogTable[n-1])
LogTable[n] = (BYTE)(LogTable[n-1]+1);
}
long far ExpFloat14(long xx) {
static float fLogTwo = (float)0.6931472;
int f;
long Ans;
f = 23 - (int)RegFloat2Fg(RegDivFloat(xx, *(long*)&fLogTwo), 0);
Ans = ExpFudged(RegFloat2Fg(xx, 16), f);
return(RegFg2Float(Ans, (char)f));
}
double TwoPi;
_CMPLX temp, BaseLog;
_CMPLX cdegree = { 3.0, 0.0 }, croot = { 1.0, 0.0 };
int ComplexNewtonSetup(void) {
threshold = .001;
periodicitycheck = 0;
if(param[0] != 0.0 || param[1] != 0.0 || param[2] != 0.0 ||
param[3] != 0.0) {
croot.x = param[2];
croot.y = param[3];
cdegree.x = param[0];
cdegree.y = param[1];
FPUcplxlog(&croot, &BaseLog);
TwoPi = asin(1.0) * 4;
}
return(1);
}
int ComplexNewton(void) {
_CMPLX cd1;
/* new = ((cdegree-1) * old**cdegree) + croot
----------------------------------
cdegree * old**(cdegree-1) */
cd1.x = cdegree.x - 1.0;
cd1.y = cdegree.y;
temp = ComplexPower(old, cd1);
FPUcplxmul(&temp, &old, &new);
tmp.x = new.x - croot.x;
tmp.y = new.y - croot.y;
if((sqr(tmp.x) + sqr(tmp.y)) < threshold)
return(1);
FPUcplxmul(&new, &cd1, &tmp);
tmp.x += croot.x;
tmp.y += croot.y;
FPUcplxmul(&temp, &cdegree, &cd1);
FPUcplxdiv(&tmp, &cd1, &old);
if(DivideOverflow)
{
DivideOverflow = 0;
return(1);
}
new = old;
return(0);
}
int ComplexBasin(void) {
_CMPLX cd1;
double mod;
/* new = ((cdegree-1) * old**cdegree) + croot
----------------------------------
cdegree * old**(cdegree-1) */
cd1.x = cdegree.x - 1.0;
cd1.y = cdegree.y;
temp = ComplexPower(old, cd1);
FPUcplxmul(&temp, &old, &new);
tmp.x = new.x - croot.x;
tmp.y = new.y - croot.y;
if((sqr(tmp.x) + sqr(tmp.y)) < threshold) {
if(fabs(old.y) < .01)
old.y = 0.0;
FPUcplxlog(&old, &temp);
FPUcplxmul(&temp, &cdegree, &tmp);
mod = tmp.y/TwoPi;
coloriter = (long)mod;
if(fabs(mod - coloriter) > 0.5) {
if(mod < 0.0)
coloriter--;
else
coloriter++;
}
coloriter += 2;
if(coloriter < 0)
coloriter += 128;
return(1);
}
FPUcplxmul(&new, &cd1, &tmp);
tmp.x += croot.x;
tmp.y += croot.y;
FPUcplxmul(&temp, &cdegree, &cd1);
FPUcplxdiv(&tmp, &cd1, &old);
if(DivideOverflow)
{
DivideOverflow = 0;
return(1);
}
new = old;
return(0);
}
/*
* Generate a gaussian distributed number.
* The right half of the distribution is folded onto the lower half.
* That is, the curve slopes up to the peak and then drops to 0.
* The larger slope is, the smaller the standard deviation.
* The values vary from 0+offset to range+offset, with the peak
* at range+offset.
* To make this more complicated, you only have a
* 1 in Distribution*(1-Probability/Range*con)+1 chance of getting a
* Gaussian; otherwise you just get offset.
*/
int GausianNumber(int Probability, int Range) {
int n, r;
long Accum = 0, p;
p = divide((long)Probability << 16, (long)Range << 16, 16);
p = multiply(p, con, 16);
p = multiply((long)Distribution << 16, p, 16);
if(!(rand15() % (Distribution - (int)(p >> 16) + 1))) {
for(n = 0; n < Slope; n++)
Accum += rand15();
Accum /= Slope;
r = (int)(multiply((long)Range << 15, Accum, 15) >> 14);
r = r - Range;
if(r < 0)
r = -r;
return(Range - r + Offset);
}
return(Offset);
}
#endif