Fsexp.cin

// fsexp.cin is routine for the fast evaluation of natural tetration

// Specification: // complex<double>FSEXP( complex <double> z) //Other functions defined below provide approximations for various parts of the complex plane.

/*
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex.h>
#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
*/

z_type fima(z_type z){ z_type c,e;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
//z_type R=z_type(1.0779614375278 , -.9465409639480); 
//z_type R=z_type(1.0779614375278 , -.9465409639479); 
  z_type R=z_type(1.0779614375280 , -.9465409639480); 
z_type a2=.5/(Zo-1.);
z_type a3=(a2+1./6.)/(Zo*Zo-1.);
z_type a4=(a2/2.+a2*a2/2.+a3+1./24.)/(Zo*Zo*Zo-1.);
z_type a5=(.5*a2*a2+a2/6.+a2*a3+a3/2.+a4+1./120.)/(Zo*Zo*Zo*Zo-1.);
z_type Li=2*M_PI*I;
z_type b0=z_type(0.1223,       -0.02370);
//printf("b0=%11.5e %11.5e\n", Re(b0),Im(b0 )) ;
e=exp(Zo*z+R); 
c= Zo + e*( 1. + e*(a2+e*(a3+e*(a4+e*a5))) + b0*exp(Li*z) );
return c;}
//#include "fima.cin"

z_type tai3(z_type z)
{ int K=50,k;
z_type DER3[51]={
 z_type( 0.37090658903228507226, 1.33682167078891400713)
,z_type( 0.03660096537598455518, 0.13922215389950498565)
,z_type(-0.16888431840641535131, 0.09718533619629270148)
,z_type(-0.12681315048680869007,-0.11831628767028627702)
,z_type( 0.04235809310323926380,-0.10520930088320722129)
,z_type( 0.05848306393563178218,-0.00810224524496080435)
,z_type( 0.02340031665294847393, 0.01807777011820375229)
,z_type( 0.00344260984701375092, 0.01815103755635914459)
,z_type(-0.00803695814441672193, 0.00917428467034995393)
,z_type(-0.00704695528168774229,-0.00093958506727472686)
,z_type(-0.00184617963095305509,-0.00322342583181676459)
,z_type( 0.00054064885443097391,-0.00189672061015605498)
,z_type( 0.00102243648088806748,-0.00055968657179243165)
,z_type( 0.00064714396398048754, 0.00025980661935827123)
,z_type( 0.00010444455593372213, 0.00037199472598828116)
,z_type(-0.00011178535404343476, 0.00016786687552190863)
,z_type(-0.00010630158710808594, 0.00002072200033125881)
,z_type(-0.00005078098819110608,-0.00003575913005741248)
,z_type(-0.00000314742998690270,-0.00003523185937587781)
,z_type( 0.00001347661344130504,-0.00001333034137448205)
,z_type( 0.00000980239082395275, 0.00000047607184151673)
,z_type( 0.00000355493475454698, 0.00000389816212201278)
,z_type(-0.00000021552652645735, 0.00000296273413237997)
,z_type(-0.00000131673903627820, 0.00000097381354534333)
,z_type(-0.00000083401960806066,-0.00000018663858711081)
,z_type(-0.00000022869610981361,-0.00000037497716770031)
,z_type( 0.00000005372584613379,-0.00000023060136585176)
,z_type( 0.00000011406656653786,-0.00000006569510293486)
,z_type( 0.00000006663595460757, 0.00000002326630571343)
,z_type( 0.00000001396786846375, 0.00000003315118300198)
,z_type(-0.00000000684890556421, 0.00000001713041981611)
,z_type(-0.00000000916619598268, 0.00000000403886083652)
,z_type(-0.00000000502933384276,-0.00000000222121299478)
,z_type(-0.00000000084484352792,-0.00000000273668661113)
,z_type( 0.00000000070086729861,-0.00000000124687683156)
,z_type( 0.00000000070558101710,-0.00000000021962577544)
,z_type( 0.00000000035900951951, 0.00000000018774741308)
,z_type( 0.00000000005248658571, 0.00000000021201177126)
,z_type(-0.00000000006264758835, 0.00000000009059171879)
,z_type(-0.00000000005333473585, 0.00000000001006078866)
,z_type(-0.00000000002432138144,-0.00000000001506937008)
,z_type(-0.00000000000331880379,-0.00000000001544700067)
,z_type( 0.00000000000501652570,-0.00000000000658967459)
,z_type( 0.00000000000401214135,-0.00000000000036708383)
,z_type( 0.00000000000158629111, 0.00000000000119885992)
,z_type( 0.00000000000019668766, 0.00000000000106532662)
,z_type(-0.00000000000036355730, 0.00000000000047229527)
,z_type(-0.00000000000029920206, 0.00000000000001251827)
,z_type(-0.00000000000010305550,-0.00000000000009571381)
,z_type(-0.00000000000000910369,-0.00000000000007087680)
,z_type( 0.00000000000002418310,-0.00000000000003240337)
};
//#include "Tai3.inc"
z_type s=0.,t=1.;
 z-=z_type(0.,3.); z/=2.; 
for(k=0;k<K;k++) { s+=DER3[k]*t; t*=z; }
return s;
}

z_type maclo(z_type z) {  int K=100,k;
DB d[110]={
  0.30685281944005469058 
, 1.18353470251664338875
, 1.58593285160678321155
, 1.36629265207672068172
, 1.36264601823980036066
, 1.21734246689515424045
, 1.10981816083559525765
, 0.96674692974769849130
, 0.84089872598668435888
, 0.71353210966804747617
, 0.60168548504001373445
, 0.49928574281440518678
, 0.41140086629121763728
, 0.33506195665178500898
, 0.27104779243942234146
, 0.21728554054610033086
, 0.17311050207880035456
, 0.13690016038526570119
, 0.10765949732729711286
, 0.08413804539743192923
, 0.06542450487497340761
, 0.05060001212013485322
, 0.03895655493977817629
, 0.02985084640296329153
, 0.02277908979501017117
, 0.01730960309240666892
, 0.01310389615589767874
, 0.00988251130733762764
, 0.00742735935367278347
, 0.00556296426263720549
, 0.00415334478103463346
, 0.00309116153137843543
, 0.00229387529664008653
, 0.00169729976398295653
, 0.00125245885041635465
, 0.00092172809095368547
, 0.00067661152429638357
, 0.00049544127485341987
, 0.00036192128589181518
, 0.00026376927786672476
, 0.00019180840045267570
, 0.00013917553105723647
, 0.00010077412023867018
, 0.00007281884753121133
, 0.00005251474516228446
, 0.00003779882770351268
, 0.00002715594536867241
, 0.00001947408515177282
, 0.00001394059355016322
, 0.00000996213949015693
, 0.00000710713872292710
, 0.00000506199803708578
, 0.00000359960968975399
, 0.00000255569149787694
, 0.00000181175810338313
, 0.00000128245831538430
, 0.00000090647322737496
, 0.00000063980422418981
, 0.00000045095738191441
, 0.00000031741772125007
, 0.00000022312521183625
, 0.00000015663840476155
, 0.00000010982301013230
, 0.00000007690305934973
, 0.00000005378502675604
, 0.00000003757126131521
, 0.00000002621429405247
, 0.00000001826909956818
, 0.00000001271754463425
, 0.00000000884310192977
, 0.00000000614230041407
, 0.00000000426177146865
, 0.00000000295386817285
, 0.00000000204522503591
, 0.00000000141464900426
, 0.00000000097750884878
, 0.00000000067478454029
, 0.00000000046535930671
, 0.00000000032062550784
, 0.00000000022069891976
, 0.00000000015177557961
, 0.00000000010428189463
, 0.00000000007158597119
, 0.00000000004909806710
, 0.00000000003364531769
, 0.00000000002303635851
, 0.00000000001575933679
, 0.00000000001077213757
, 0.00000000000735717912
, 0.00000000000502077719
, 0.00000000000342362421
, 0.00000000000233271256
, 0.00000000000158818623
, 0.00000000000108046566
, 0.00000000000073450488
, 0.00000000000049894945
, 0.00000000000033868911
, 0.00000000000022973789
, 0.00000000000015572383
, 0.00000000000010548054
, 0.00000000000007139840
, 0.00000000000004829557
, 0.00000000000003264619
, 0.00000000000002205299
, 0.00000000000001488731
, 0.00000000000001004347
, 0.00000000000000677124
, 0.00000000000000456225
, 0.00000000000000307196
, 0.00000000000000206720
};

z_type s=0.;
z_type z2=z/2.;
z_type t=1.; 
for(k=0;k<=K;k++) { s+=d[k]*t; t*=z2; }
return s+log(z+2.);
}
//#include "maclo.cin"

//#include "f4natu.cin"
z_type  FIMA(z_type z){        DB x=Re(z);
                       DB y=Im(z);
                       if(y < 0.2379*x) return exp(FIMA(z-1.));
                       return fima(z); }
z_type TAI3(z_type z){ DB x=Re(z);
                       if(x > 0.5) return exp(TAI3(z-1.));
                       if(x < -.5) return log(TAI3(z+1.));
                       return tai3(z); }

z_type MACLO(z_type z){        DB x=Re(z); z_type c;
                       //if(x > 0.5 && x<3.66)
                       if(x > 0.5)
                               {       c=z-1.;
                                       c=MACLO(c);
                                       c=exp(c);
                               //      printf("about to return %9.3e %9.3e\n",Re(c),Im(c));
                               //      getchar();
                                       return c;
                               }
                       if(x < -.5) return log(MACLO(z+1.));
                       return maclo(z); }

z_type FSEXP(z_type z){DB y=Im(z);
if(y> 4.5) return FIMA(z);
if(y> 1.5) return TAI3(z);
if(y>-1.5) return MACLO(z);
if(y>-4.5) return conj(TAI3(conj(z)));
           return conj(FIMA(conj(z)));
}

Keywords

Natural tetration, tetration

I expected to make it half shorter (and even faster) with the same precision, but I had no time to finish it.. So, I upload that I have. I believe, the professional programmers can do it better. The algorithm is described in the reference below. Use for free, attribute the source. Kouznetsov 19:02, 1 March 2012 (JST)

http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.