Fooling with pslq (again) In[1209]:= 3 ArcTan[1/7] + 2 ArcTan[2/11] // FullSimplify (or ArcTan[1/5.5]) Out[1209]=π/4 Holy cow! Isn't that slightly better (at least for a true binary computation) than In[1208]:= 4 ArcTan[1/5] - ArcTan[1/239] // FullSimplify Out[1208]= π/4 with which all the old decimalization records were set? Then I stumbled upon In[1210]:= 5 ArcTan[1/7] + 2 ArcTan[3/79] // FullSimplify Out[1210]=π/4 (in http://www.cs.utsa.edu/~wagner/pi/ruby/pi_atan.html ) Euler was inhuman. I suspect the adherence to Machin was because, even on binary machines, people did the whole computation base 10^10 to avoid a big conversion later. Decimal. Feh. --rwg