AuFac.cin

// AuFac.cin defines z_type AuFac(z_type) that evaluates the AuFac, id est, (ArcSuperFactorial), id est, the Abel function of Factorial. In wide ranges of values of z, \(\rm AuFac(Sufac(z))\!=\!z~\) and \(~\rm SuFac(Aufac(z))\!=\!z~\). // Of order of 14 correct decimal digits are expected in the result.

z_type arcsuperfac0(z_type z){ int n; z_type s, c, e;
DB k=0.61278745233070836381366079016859252;
DB U[19]={1.,                           -0.798731835172434541585621072345730147,
0.69806411355936704552792746483306691,  - 0.6339640557572814865638000833478131,   
0.5884152357911398848274232132172143,   -0.5538887519936519511632593654732843,  
0.526547902598592454703287733600892,    -0.504191460428021561516069870422848,   
0.48545298002933922263549078734881,     -0.46943468090947139273094056497701,
0.4555204862393622788179080677150,      -0.4432726222110411295132308010077,     
0.4323708863150174727399798603985,      -0.4225752531177612936293974175008,     
0.413701949171132722406449918702,       -0.40560764595293667778491699902, 
0.39817872478532299454624349817,        -0.391323       -0.384};
//      z-=2.; s=U[15]*z; for(n=14;n>=0;n--){ s+=U[n]; s*=z;}
       z-=2.; s=U[18]*z; for(n=17;n>=0;n--){ s+=U[n]; s*=z;}
return log(s)/k;}
z_type AuFac(z_type z){ if(abs(z-2.)<.12) return arcsuperfac0(z);
                               return  AuFac(afacc(z))+1.;}

//