F45L.cin

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

z_type f45L(z_type z){ int n; z_type e,s;
DB Uco[21]={1,  
-.44858743119526122890,        .21208912005491969757, 
-.10218436750697167872,        0.49698683037371830337e-1, 
-0.2430759032611955221e-1,     0.11933088396510860210e-1,
-0.587369764200886206e-2,      0.289686728710575713e-2,
-0.1430908106079253664e-2,     0.7076637148565759223e-3,
-0.3503296158729878e-3,        0.17357560046634138e-3,
-0.86061011929162626e-4,       0.426959089012891e-4,
-0.2119302906819844809e-4,     0.1052442259960e-4,
-0.52285174354086e-5,          0.259844999161e-5,
-0.129178211214818578e-5,      0.4e-6  };
       z-=4.;
       s=Uco[19]; for(n=18; n>=0; n--){ s*=z; s+=Uco[n]; }
//     s=Uco[20]; for(n=19; n>=0; n--){ s*=z; s+=Uco[n]; }
//      return log(s*z)/.32663425997828098238 +1.1152091357215375; 
       return log(s*z)/.32663425997828098238 +1.11520724513161;
 }

z_type F45L(z_type z){ DB b=sqrt(2.);
                       if(abs(z-4.)>.4) return F45L(log(z)/log(b))+1. ;
                        return f45L(z);
               }

/*

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.

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