[math-fun] We have noted that Gene's x+sin x HAKMEM recurrence

converges cubically to odd multiples of π. E.g., In[539]:= Series[x+Sin@x,{x,3π,3}] Out[539]= 3 π+1/6 (x-3 π)^3+O[x-3π]^4 Rohan notes that x-sin x converges cubically to the even multiples: In[523]:= Series[x-Sin@x,{x,2π,3}] Out[523]= 2 π+1/6 (x-2 π)^3+O[x-2 π]^4 and x+cos x converges cubically to (2n+1/2)π, In[519]:= Series[Cos[x]+x,{x,5π/2,3}] Out[519]= 5 π/2+1/6 (x-(5π)/2)^3+O[x-5 π/2]^4 In[540]:= Series[Cos[x]+x,{x,π/2,3}] Out[540]=π/2+1/6 (x-π/2)^3+O[x-π/2]^4 and x-cos x converges to (2n+3/2)π: In[522]:= Series[x-Cos[x],{x,3π/2,3}] Out[522]= (3π)/2+1/6 (x-(3π)/2)^3+O[x-(3π)/2]^4 --rwg "... given the difficulty related to the calculation of sinuses." Indeed few things are as difficult as pushing stones up your nose.