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// analuxpf3c.cin is C++ implementation of the elementary function \(\mathrm{Fit}_3\) and related functions, that provides approximation of natural tetration for moderate values of the argument.

// This fit had been used to guess the asymptotic behaviour of tetration at \(\mathrm i \infty\).

// z_type should be defined as complex <double>

// In order to evaluate the approximation at artument \(z\), type F3(z)


z_type s2(z_type x){ return exp(exp(x-2.51))-.6+.08*(x+1.); }
z_type f1(z_type x){ return log(x+2.) + exp(x-1.);}
z_type f2(z_type x){ return log(2.+x) + (1.+x)*(
        1. + .5*exp((x-1.)*s2(x))*x*(M_E-2.+log(4./3.)) - log(2.)
z_type f3(z_type x){ return .6*f2(x)+.4*log(f2(x+1.)) ;}
z_type F3(z_type z){ DB x=Re(z);
if(x<-1) return log(F3(z+1.));
if(x>0) return exp(F3(z-1.));
return f3(z);
z_type F2(z_type z){ DB x=Re(z);
if(x<-1) return log(F2(z+1.));
if(x>0) return exp(F2(z-1.));
return f2(z);
z_type F1(z_type z){ DB x=Re(z);
if(x<-1) return log(F1(z+1.));
if(x>0) return exp(F1(z-1.));
return f1(z);

References D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.