LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(y-\mu)^2}{2\sigma^2} \right) h(y) dy}

${\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(y-\mu )^{2}}{2\sigma ^{2}}}\right)h(y)dy}$

... is translated to the CAS output ...

Semantic latex: E [h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sigma \sqrt{2 \cpi}} \exp(- \frac{(y-\mu)^2}{2\sigma^2}) h(y) \diff{y}

Confidence: 0

Mathematica

Translation: E[h[y]] == Integrate[Divide[1,\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \[Mu])^(2),2*\[Sigma]^(2)]]*h[y], {y, - Infinity, + Infinity}, GenerateConditions->None]

Information

Sub Equations

E[h[y]] = Integrate[Divide[1,\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \[Mu])^(2),2*\[Sigma]^(2)]]*h[y], {y, - Infinity, + Infinity}, GenerateConditions->None]

Free variables

\[Mu]

\[Sigma]

y

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Exponential function; Example: \exp@@{z}

Will be translated to: Exp[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Mathematica: https://reference.wolfram.com/language/ref/Exp.html

Pi was translated to: Pi

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Test expression: (E*(h*(y)))-(Integrate[Divide[1,\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \[Mu])^(2),2*\[Sigma]^(2)]]*h*(y), {y, - Infinity, + Infinity}, GenerateConditions->None])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Numeric

SymPy

Translation: E(h(y)) == integrate((1)/(Symbol('sigma')*sqrt(2*pi))*exp(-((y - Symbol('mu'))**(2))/(2*(Symbol('sigma'))**(2)))*h(y), (y, - oo, + oo))

Information

Sub Equations

E(h(y)) = integrate((1)/(Symbol('sigma')*sqrt(2*pi))*exp(-((y - Symbol('mu'))**(2))/(2*(Symbol('sigma'))**(2)))*h(y), (y, - oo, + oo))

Free variables

Symbol('mu')

Symbol('sigma')

y

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Exponential function; Example: \exp@@{z}

Will be translated to: exp($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 SymPy: https://docs.sympy.org/latest/modules/functions/elementary.html#exp

Pi was translated to: pi

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Maple

Translation: E(h(y)) = int((1)/(sigma*sqrt(2*Pi))*exp(-((y - mu)^(2))/(2*(sigma)^(2)))*h(y), y = - infinity..+ infinity)

Information

Sub Equations

E(h(y)) = int((1)/(sigma*sqrt(2*Pi))*exp(-((y - mu)^(2))/(2*(sigma)^(2)))*h(y), y = - infinity..+ infinity)

Free variables

mu

sigma

y

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Exponential function; Example: \exp@@{z}

Will be translated to: exp($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace

Pi was translated to: Pi

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$y$

$h$

Description

integral

integration by substitution

expectation

variable

Hermite polynomial

Complete translation information:

{
  "id" : "FORMULA_1423fe2a603a0984f3b824f4dd88eeaa",
  "formula" : "E[h(y)] = \\int_{-\\infty}^{+\\infty} \\frac{1}{\\sigma \\sqrt{2\\pi}} \\exp \\left( -\\frac{(y-\\mu)^2}{2\\sigma^2} \\right) h(y) dy",
  "semanticFormula" : "E [h(y)] = \\int_{-\\infty}^{+\\infty} \\frac{1}{\\sigma \\sqrt{2 \\cpi}} \\exp(- \\frac{(y-\\mu)^2}{2\\sigma^2}) h(y) \\diff{y}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "E[h[y]] == Integrate[Divide[1,\\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \\[Mu])^(2),2*\\[Sigma]^(2)]]*h[y], {y, - Infinity, + Infinity}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "E[h[y]] = Integrate[Divide[1,\\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \\[Mu])^(2),2*\\[Sigma]^(2)]]*h[y], {y, - Infinity, + Infinity}, GenerateConditions->None]" ],
        "freeVariables" : [ "\\[Mu]", "\\[Sigma]", "y" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\exp" : "Exponential function; Example: \\exp@@{z}\nWill be translated to: Exp[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.2#E19\nMathematica:  https://reference.wolfram.com/language/ref/Exp.html",
          "\\cpi" : "Pi was translated to: Pi",
          "E" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "E*(h*(y))",
          "rhs" : "Integrate[Divide[1,\\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \\[Mu])^(2),2*\\[Sigma]^(2)]]*h*(y), {y, - Infinity, + Infinity}, GenerateConditions->None]",
          "testExpression" : "(E*(h*(y)))-(Integrate[Divide[1,\\[Sigma]*Sqrt[2*Pi]]*Exp[-Divide[(y - \\[Mu])^(2),2*\\[Sigma]^(2)]]*h*(y), {y, - Infinity, + Infinity}, GenerateConditions->None])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "E(h(y)) == integrate((1)/(Symbol('sigma')*sqrt(2*pi))*exp(-((y - Symbol('mu'))**(2))/(2*(Symbol('sigma'))**(2)))*h(y), (y, - oo, + oo))",
      "translationInformation" : {
        "subEquations" : [ "E(h(y)) = integrate((1)/(Symbol('sigma')*sqrt(2*pi))*exp(-((y - Symbol('mu'))**(2))/(2*(Symbol('sigma'))**(2)))*h(y), (y, - oo, + oo))" ],
        "freeVariables" : [ "Symbol('mu')", "Symbol('sigma')", "y" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\exp" : "Exponential function; Example: \\exp@@{z}\nWill be translated to: exp($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.2#E19\nSymPy: https://docs.sympy.org/latest/modules/functions/elementary.html#exp",
          "\\cpi" : "Pi was translated to: pi",
          "E" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "E(h(y)) = int((1)/(sigma*sqrt(2*Pi))*exp(-((y - mu)^(2))/(2*(sigma)^(2)))*h(y), y = - infinity..+ infinity)",
      "translationInformation" : {
        "subEquations" : [ "E(h(y)) = int((1)/(sigma*sqrt(2*Pi))*exp(-((y - mu)^(2))/(2*(sigma)^(2)))*h(y), y = - infinity..+ infinity)" ],
        "freeVariables" : [ "mu", "sigma", "y" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\exp" : "Exponential function; Example: \\exp@@{z}\nWill be translated to: exp($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.2#E19\nMaple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace",
          "\\cpi" : "Pi was translated to: Pi",
          "E" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 1,
    "word" : 10
  } ],
  "includes" : [ "y", "h" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "integral",
    "score" : 0.7125985104912714
  }, {
    "definition" : "integration by substitution",
    "score" : 0.6859086196238077
  }, {
    "definition" : "expectation",
    "score" : 0.6460746792928004
  }, {
    "definition" : "variable",
    "score" : 0.6460746792928004
  }, {
    "definition" : "Hermite polynomial",
    "score" : 0.5988174995334326
  } ]
}

Specify your own input

Formula input
Wikitext context	{{short description\|Form of Gaussian quadrature}} [[File:Gauss-Hermite quadrature weights.svg\|thumb\|Weights versus ''x<sub>i</sub>'' for four choices of ''n'']] In [[numerical analysis]], '''Gauss–Hermite quadrature''' is a form of [[Gaussian quadrature]] for approximating the value of integrals of the following kind: :<math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx.</math> In this case :<math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</math> where ''n'' is the number of sample points used. The ''x''<sub>''i''</sub> are the roots of the physicists' version of the [[Hermite polynomial]] ''H''<sub>''n''</sub>(''x'') (''i'' = 1,2,...,''n''), and the associated weights ''w''<sub>''i''</sub> are given by <ref name="AS">[[Abramowitz and Stegun\|Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'']], 10th printing with corrections (1972), Dover, {{ISBN\|978-0-486-61272-0}}. Equation 25.4.46.</ref> :<math>w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.</math> ==Example with change of variable== Consider a function ''h(y)'', where the variable ''y'' is [[Normal distribution\|Normally distributed]]: <math> y \sim \mathcal{N}(\mu,\sigma^2)</math>. The [[Expected value\|expectation]] of ''h'' corresponds to the following integral: <math>E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(y-\mu)^2}{2\sigma^2} \right) h(y) dy</math> As this doesn't exactly correspond to the Hermite polynomial, we need to change variables: <math>x = \frac{y-\mu}{\sqrt{2} \sigma} \Leftrightarrow y = \sqrt{2} \sigma x + \mu</math> Coupled with the [[integration by substitution]], we obtain: <math>E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sqrt{\pi}} \exp(-x^2) h(\sqrt{2} \sigma x + \mu) dx</math> leading to: <math>E[h(y)] \approx \frac{1}{\sqrt{\pi}} \sum_{i=1}^n w_i h(\sqrt{2} \sigma x_i + \mu)</math> ==References== <references/> * {{dlmf\| id = 3.5.E28\| title=Quadrature: Gauss–Hermite Formula}} * {{cite journal\|first1=T. S. \| last1=Shao \| first2=T. C. \| last2=Chen \| first3= R. M. \| last3=Frank \|title=Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials \|year=1964 \|journal=Math. Comp. \| mr=0166397 \| volume=18 \| number=88 \| pages=598–616 \|doi=10.1090/S0025-5718-1964-0166397-1\| doi-access=free }} * {{cite journal\|first1=N. M. \| last1=Steen \| first2=G. D. \| last2=Byrne \|first3=E. M. \| last3=Gelbard \| title = Gaussian quadratures for the integrals <math>\textstyle\int_0^\infty e^{-x^2} f(x) dx</math> and <math>\textstyle\int_0^b e^{-x^2} f(x) dx</math> \|journal = Math. Comp. \| year=1969 \|volume=23 \| number=107 \| pages=661–671 \| mr=0247744 \| doi=10.1090/S0025-5718-1969-0247744-3 \| doi-access=free }} * {{cite journal\|first1= B. \| last1=Shizgal \| title = A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems \|journal=J. Comput. Phys. \| volume=41 \| pages=309–328 \| year=1981 \|doi=10.1016/0021-9991(81)90099-1 }} ==External links== * For tables of Gauss-Hermite abscissae and weights up to order ''n'' = 32 see http://www.efunda.com/math/num_integration/findgausshermite.cfm. * [http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html Generalized Gauss–Hermite quadrature], [[free software]] in C++, Fortran, and Matlab {{DEFAULTSORT:Gauss-Hermite quadrature}} [[Category:Numerical integration (quadrature)]]
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LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

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