Naively we want to minimize -sum 1/log x and by that metric this is worse than Machin: -1/log(1/7) - 1/log(239/4802) = 0.847... -1/log(1/5) - 1/log(1/239) = 0.803... It only gets worse if you expand the model to make unit fractions less costly. (Exercise: what's the right discount here? Depends on the relative speed of "divide by a small integer" and "add two multiprecision numbers".) But I guess it beats it in the sense that 1/7 < 1/5? Charles Greathouse Analyst/Programmer Case Western Reserve University On Thu, Sep 25, 2014 at 5:38 AM, Bill Gosper <billgosper@gmail.com> wrote:
Doesn't this, in some sense, beat Machin? In[628]:= TrigExpand[Sin[4*ArcSin[1/7] - ArcSin[239/4802]]] Out[628]= 1/2 Is it trying to tell us something?
rcs>The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239). ("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?) Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification. Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:
These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]] Out[526]= 1/2 In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]] Out[528]= 1/2 In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]] Out[532]= 1/2 In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]] Out[517]= 1/2 I.e., ArcSin[97/961] + 2*ArcSin[13/62] = π/6, but Wolfram's nascent artificial intelligence is too stupid to notice. I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf). --rwg
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