F45E.cin

// F45E.cin is routine for evaluation of the superfunction of the exponential to base $b\!=\!\sqrt{2}$, constructed at the fixed point $L\!=\!4$.

z_type f45E(z_type z){int n; z_type e,s;
DB coeu[21]={1.,
0.44858743119526122890, .19037224679780675668, 
0.77829576536968278770e-1, 0.30935860305707997953e-1, 
0.12022125769065893274e-1, 0.45849888965617461424e-2,
0.17207423310577291102e-2, 0.63681090387985537364e-3,
0.23276960030302567773e-3, 0.84145511838119915857e-4,
0.30115646493706434120e-4, 0.10680745813035087964e-4,
0.37565713615564248453e-5, 0.13111367785052622794e-5,
0.45437916254218158081e-6, 0.15642984632975371803e-6,
0.53523276400816416929e-7, 0.18207786280204973113e-7,  
0.61604764947389226744e-8, 0.2e-8};
       e=exp(.32663425997828098238*(z-1.11520724513161));      
       s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } 
       return 4.+s*e;}

z_type F45E(z_type z){ DB b=sqrt(2.);
                       if(Re(z)<-1.) return f45E(z);
                       return exp(F45E(z-1.)*log(b));
               }

/*

References

http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.

• /