/*  sdfcosrn.cpp  by K.Tsuru  */
// function ID 3205     DRADIX
/******************************************************************
SDouble class
trigonometric function cos x using x/R^n reduction method
[Algorithm]
Pay attention to a formura
    cos(4*x) = 8*cos(x)^4 - 8*cos(x)^2 + 1.
step 1 : Let R = 4 and d = x/(R^n)(n is an integer).
step 2 : Evaluate y(n) = cos d by series.
step 3 : While n > 0, repeat the recurrence formula
    y(n-1) <-- 8*y(n)^4 - 8*y(n)^2 + 1,  n <-- n-1.
It is better to adjust the value of n examining the effective figures.
When n is too large,it becomes slower.
Using the conbination with a formula
    cos(2*x)=2*cos(x)^2-1
and taking R=8,it becomes faster a little.
*******************************************************************/
#ifndef SN_H
#include "sn.h"
#endif
SDouble CosRN(const SDouble& x){  // |x| <= pi/4
    int k;
    const uint R = 8u;
    const double logR = log10((double)R);   // ver. 2.17
    uint effFig = x.EffFig(), upPrec;
    double pw = log10((double)effFig);
    const int n = int(0.5*DFIGURES*pw*pw) + x.RdxExp();

    if(n <= 0) return CosSeries(x); // enough small
// n > 0
    k = int( (double)n*logR )+ 1;   //figures of R^n

    upPrec = k/DFIGURES + (uint)pw + 1u;
    if(k % DFIGURES) upPrec++;

    RealSize C;
    SDouble y, u;
    uint up = x.ProperUpPrec(upPrec);
//Raises up the precision if possible.
    if(up) C.SetEffFig(effFig + up, C.TEMP_EXTEND);
    u = x/Dpow(R, n);   // X = x/(R^n)
    y = CosSeries(u); // CosSeries(u);
    k = n;
    x.iterationCount = k;
    while(k > 0){
        //double angle formula
        y = DsMult(y*y, 2) -ONE;
        //four times angle formula
        //faster than a method using double angle formula three times.
        u = y*y;
        u = u*u-u;
        y = DsMult(u, 8) + ONE;
        k--;
    }
    if(up){
        C.SetEffFig(0);
        y.Reform(3205);
    }
    return y;
}