My Quibbles  

The Inverse Symbolic Calculator 1 and Plouffe's Inverter 2 are great resources, but they failed to satisfy me, primarily for the following reasons:

Plouffe or ISC ries
Cryptic output and no glossary (but see below). Gives answers like -1+x^5+x^6-x^7-x^8+x^9+x^11+x^12 without defining what x is. Also gives many answers that can't be checked unless you own a copy of Mathematica or Maple. All solutions are self-explanatory. Lesser-known symbols like phi are defined by a tagline in the output the first time they are used.
Presents all answers in numerical sequence and without regard for the complexity of the description of each number.1 Gives answers in order of increasing complexity and increasing closeness to your number — just like continued fractions.
Isn't useful for people who only know a number to within 6 or 7 digits, but know that it probably has a simple formula — because there are just so many long, complex formulas to wade through. Immediately gives you the simplest formulas or equations for any number regardless of the number of digits in your number. It also gives matches that are equal to your value in the first 6 or 7 digits, but you're free to just ignore those matches.
Thinks decimal points don't matter — search for 2.5061841 and it gives ln(26263/20441) = 0.25061841... as an answer! Why?!?!? Doesn't consider "a factor of 10" to be any more important than "a factor of 9" or "a factor of 11". Numbers are weighted by size and how easily they can be formed from other numbers (giving composites an advantage).
As of late 2011, still didn't know that 2.506184145588... is the solution of xx = 10 (Try it) All solutions are in the form K = expression, and apparently the Lambert W function is not included. Reports this as the 4th solution for any value within +- 0.00044 of 2.506184.
Database is cluttered with stuff like log(26263/20441) and cos(Pi*23/56) / cos(Pi*15/34). Reports things like log(441/263) only if your number is closer to that value than to the value of any simpler expression.2
Doesn't let you specify "please ignore these weird functions". Lets you specify any combination of functions, operators, and small integers to be used in a search.
Requires Internet connection; puts load on someone else's computer. Runs on your computer, but fast enough to try billions of possible solutions in less than a minute.

Okay, enough complaining. Back to ries main page

Glossary for ISC and Plouffe's Inverter

I got tired of complaining and eventually started a list of some of the constants and functions used in ISC output.


Artin's constant 0.3739558136... (see Artin's conjecture on primitive roots)


2506186020926769 = (0001) GAM(3/4)+Artin+GAM(1/24)

    {~=} 1.2254167024 + 0.3739558136 + 23.462487693


The modified Bessel function of the second kind, commonly represented by the symbol Kα(x) for some α and x. In Maple and Mathematica this is written BesselK(α,x).


6019072301972345 = (0001) BesK(1,1)

6019072301972345 = (0297) BesselK(1,1)


Synonym for BesK.


Catalan's constant 0.915965594177219...


2506144130719856 = (0001) Catalan*(sr(5)+1/2)

    {~=} 0.915965594177 * (2.2360679774998 + 0.5)


The exponential function


  |1535063009255209 = (0028) exp(x) x=3/7


The Hypergeometric function with Maple syntax


1535063009255209 = (0231) F(1,1;3/7)

1535063009255209 = (0226) F(1/3,1/3;3/7)

2506169291781055 = (0249) F(10/23;19/39;1)

2506179399171396 = (0157) F(2/11,10/11;1/2;9/10)


The first of the two Feigenbaum constants, 4.669201609102990...


4669201609102990 = (0000) Feigenbaum constant

4669201609102990 = (0001) Feig1

  |1535061093537132 = (0001) exp(-Pi)/(K(1/sr(2))-Feig1)


The Gamma function


2506186020926769 = (0001) GAM(3/4)+Artin+GAM(1/24)

    {~=} 1.2254167024 + 0.3739558136 + 23.462487693

3625609908221908 = (0001) GAM(1/4)

3625609908221908 = (0001) Pi*sr(2)/GAM(3/4)


The Gamma function


3625609908221908 = (0092) (Gamma(1/4))^n n=1


The Gamma function


3625609908221908 = (0000) GAMMA(1/4)

3625609908221908 = (0032) GAMMA(1/4)


The Complete Elliptic Integral of the first kind. In ISC the argument is squared; for example, "K(1/sqrt(2))" in ISC meansEllipticK[1/2] using the Mathematica EllipticK[] function, and is equal to 1.854074677301...

In general, ISC's "K(x)" is equal to EllipticK[x^2] inMathematica. Using PARI/GP, ISC's "K(x)" is equal to "(Pi/2) /agm(1-x,1+x)" where we are computing the elliptic integral via the arithmetic–geometric mean.


2506185982626307 = (0001) K(1/sr(2))/(sin(Pi/5)^W(1))

2085935113295952 = (0001) K(1/sr(2))^GAM(1/3)-Pi

2085941674074939 = (0001) 2/3+K(1/sr(2))^W(1)


The Madelung constant -1.747564594633...


2506183101609269 (0001) cos(Pi/12)-ln(1+sr(2))*Madelung


Renyi's Parking constant 0.7475979202534...


2506185338545284 (0001) (W(1)-Parking*Madelung)/Parking


One or more roots of a polynomial


1535063009255209 = (0276) exp(roots(-7*x^2-4*x+3))

    = exp(3/7)   (because 3/7 is a solution of -7*x^2-4*x+3=0)


Square root


2506147722382783 = (0007) sqrt((21+sqrt(17))/4)

    {~=} sqrt((21 + 4.1231056256)/4)


Square root


2506144130719856 = (0001) Catalan*(sr(5)+1/2)

    {~=} 0.915965594177 * (2.2360679774998 + 0.5)


Sum of an infinite series, as normally indicated with "big Sigma" notation


2506181684414645 = (0013) sum((2/3*n^3-3/2*n^2+47/6*n+4)/n^(n-1),n=1..inf)

2506185297338450 = (0232) sum(1/(n!*phi(5,7,3)),n=1..inf)


The Twin prime constant, 0.6601618158...


2506141791780115 = (0001) (TwinPrim+exp(sr(2)))^sin(Pi/5)

    {~=} (0.6601618158 + e^1.414213562) ^ sin(0.5877852522)


The Lambert W function


5671432904097838 = (0000) W(1)


An ISC result consisting of a polynomial in x represents a root of that polynomial (a value that x needs to have for the polynomial's value to be zero).


6019074146223354 = (0315) -2+x+2*x^2+5*x^4+x^8

    (because if x {~=} 0.6019074146223354..., then x8 + 5x4 + 2x2 + x - 2 is zero)


The Inverse Symbolic Calculator (with a 1997 version of the database) is available here.

Plouffe's Inverter was formerly here, but is now (as of mid 2013) too big to have online but is still maintained by its creator Simon Plouffe.


1 : Plouffe's Inverter has partly addressed the progressive complexity issue by sorting its output, but it continues to only report matches that agree in the digits given by the user. Also, it scores its matches purely by string length when rendered into ASCII, so for example "Riemannzero1" gets the same score as "PisotV*Feig2" and "820421/78823" regardless of which is most exceptional.

2 : Just for fun, here are the results ries finds for the natural log of successive Fibonacci ratios:

value number given to ries ries finds
ln(3/2) 0.4054651081081 1/ex = 2/3 (5th answer)
ln(5/3) 0.5108256237659 ex-1 = 2/3 (9th answer)
ln(8/5) 0.4700036292457 ex-1 = 3/5 (5th answer)
ln(13/8) 0.4855078157817 ex-1 = 5/8 (11th answer)
ln(21/13) 0.4795730802618 3/ex = 2-1/7 (11th answer)
ln(34/21) 0.4818380868927 3 ex = 5-1/7 (12th answer)
ln(55/34) 0.4809726606163 (ex-1)/3 = 1/(5-1/7) (9th answer)
ln(89/55) 0.4813031844996 (ex-2)/3 = 1/(1/7-8) (13th answer)
ln(144/89) 0.4811769298438 2/ex-1 = 1/8+1/9 (13th answer)
ln(233/144) 0.4812251539897 ex-1/4^2 = 5/9+1 (16th answer)
ln(377/233) 0.4812067338823102 1/((ex-1)/9) = (1/4+3)^2+4 (12th answer, requires level-5 search)
ln(610/377) 0.48121376971934617 9/(ex-2) = 4-(1/4+5)^2 (24th answer, requires level-5 search)
ln(987/610) 0.48121108226612465 1/(1/ex-2/3) = 2-(5-1/4)^2 (18th answer, requires level-5 search)
ln(1597/987) 0.48121210878153085 1/(ex-5/3) = 2-(5-1/4)^2 (16th answer, requires level-5 search)
ln(2584/1597)0.48121171668748003 FAILS (ries reports "ex-442 = phi-442", a false match resulting from loss of precision. Note that phi - 2584/1597 is about 1.75×10-7 and 442 is a 10-digit number.)
ln(441/263) 0.51689084326908144 2/ex+3 = (1/(3*7)+2)2 (14th answer, requires level-4 search)

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