96.2.7 Contract atan Functions

Function: atan_contract (r) ¶

The function atan_contract(r) contracts atan functions. We assume: \(|r| < {\pi\over 2}.\)

Examples:

(%i1) load(trigtools)$

1.  

   (%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
   

2.  

   (%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
   

3. Machin’s formulae

   (%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
   

4. see http://en.wikipedia.org/wiki/Machin-like_formula

   (%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