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What mathematical program or library can guess formula for decimal fraction? For example, for number 2.414213562373095 right guess is 1 + $\sqrt 2$. Are there such tools online? I cannot find one.

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    Wolfram alpha is very good for this : Just enter a decimal expansion to get various possible closed forms! – 2017-02-26
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    @S.C.B. But 2.414213562373095 is. – 2017-02-26
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    But do not expect miracles! If you enter the decimal expansion of $\large \frac{\pi^3+e^3-\sqrt{7}}{\pi^5-e^7+\sqrt{11}}$ , you won't get your expression as a closed form. But for numbers satisfying linear or algebraic dependences with great accuracy , such as in the example, it will work very well. – 2017-02-26
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    For the case, you also accept offline tools, I suggest the free PARI/GP-calculator. It can determine algebraic and linear dependencies. – 2017-02-26
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    Using continued fractions, you can find very good rational approximations even with a table calculator. For example , $\pi$ is very close to $\frac{355}{113}$ – 2017-02-26

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Using keywords from Peter's comment ("closed form by decimal expansion") I found Inverse Symbolic Calculator Also see Sympy number identification

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    I did not know that this calculator still/again exists, but it is also very good and powerful. – 2017-02-26