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The function atan_contract(r) contracts atan functions. We assume: \(|r| < {\pi\over 2}.\)
Examples:
(%i1) load(trigtools)$
(%i2) atan_contract(atan(x)+atan(y));
(%o2)                          atan(y) + atan(x)
(%i3) assume(abs(atan(x)+atan(y))<%pi/2)$
(%i4) atan(x)+atan(y)=atan_contract(atan(x)+atan(y));
                                                 y + x
(%o4)                  atan(y) + atan(x) = atan(-------)
                                                1 - x y
(%i5) atan(1/3)+atan(1/5)+atan(1/7)+atan(1/8)$ %=atan_contract(%);
                       1         1         1         1    %pi
(%o6)             atan(-) + atan(-) + atan(-) + atan(-) = ---
                       3         5         7         8     4
(%i7) 4*atan(1/5)-atan(1/239)=atan_contract(4*atan(1/5)-atan(1/239));
                                 1          1     %pi
(%o7)                     4 atan(-) - atan(---) = ---
                                 5         239     4
(%i8) 12*atan(1/49)+32*atan(1/57)-5*atan(1/239)+12*atan(1/110443)$
%=atan_contract(%);
                1             1             1               1       %pi
(%o9)   12 atan(--) + 32 atan(--) - 5 atan(---) + 12 atan(------) = ---
                49            57           239            110443     4