Fsexp.cin

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

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

/* */
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x
 * 7) define z_type complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) 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.