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 \frac{\Gamma^2\left(\frac{1}{4}\right)}{4\sqrt{\pi}}}

${\displaystyle {\frac {\Gamma ^{2}\left({\frac {1}{4}}\right)}{4{\sqrt {\pi }}}}}$

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

Semantic latex: \frac{\EulerGamma@{z}^2(\frac{1}{4})}{4 \sqrt{\cpi}}

Confidence: 0.4934855405634

Mathematica

Translation: Divide[(Gamma[z])^(2)*(Divide[1,4]),4*Sqrt[Pi]]

Information

Sub Equations

Divide[(Gamma[z])^(2)*(Divide[1,4]),4*Sqrt[Pi]]

Free variables

z

Symbol info

Pi was translated to: Pi

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Gamma.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \EulerGamma [\EulerGamma]

Tests

Symbolic

Numeric

Maple

Translation: ((GAMMA(z))^(2)*((1)/(4)))/(4*sqrt(Pi))

Information

Sub Equations

((GAMMA(z))^(2)*((1)/(4)))/(4*sqrt(Pi))

Free variables

z

Symbol info

Pi was translated to: Pi

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\Gamma$

Description

gamma function

lemniscatic case

minimal half period

Complete translation information:

{
  "id" : "FORMULA_a7d95f86a334734d10c6c93d3d2864c6",
  "formula" : "\\frac{\\Gamma^2\\left(\\frac{1}{4}\\right)}{4\\sqrt{\\pi}}",
  "semanticFormula" : "\\frac{\\EulerGamma@{z}^2(\\frac{1}{4})}{4 \\sqrt{\\cpi}}",
  "confidence" : 0.49348554056339716,
  "translations" : {
    "Mathematica" : {
      "translation" : "Divide[(Gamma[z])^(2)*(Divide[1,4]),4*Sqrt[Pi]]",
      "translationInformation" : {
        "subEquations" : [ "Divide[(Gamma[z])^(2)*(Divide[1,4]),4*Sqrt[Pi]]" ],
        "freeVariables" : [ "z" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/5.2#E1\nMathematica:  https://reference.wolfram.com/language/ref/Gamma.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\EulerGamma [\\EulerGamma]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "((GAMMA(z))^(2)*((1)/(4)))/(4*sqrt(Pi))",
      "translationInformation" : {
        "subEquations" : [ "((GAMMA(z))^(2)*((1)/(4)))/(4*sqrt(Pi))" ],
        "freeVariables" : [ "z" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/5.2#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ {
    "section" : 0,
    "sentence" : 2,
    "word" : 15
  } ],
  "includes" : [ "\\Gamma" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "gamma function",
    "score" : 0.6859086196238077
  }, {
    "definition" : "lemniscatic case",
    "score" : 0.6859086196238077
  }, {
    "definition" : "minimal half period",
    "score" : 0.6859086196238077
  } ]
}

Specify your own input

Formula input
Wikitext context	{{Expand German\|Lemniskatischer Sinus\|date=August 2017}} In [[mathematics]], a '''lemniscatic elliptic function''' is an [[elliptic function]] related to the arc length of a [[lemniscate of Bernoulli]] studied by [[Giulio Carlo de' Toschi di Fagnano]] in 1718. It has a square period lattice and is closely related to the [[Weierstrass elliptic function]] when the Weierstrass invariants satisfy {{math\|''g''<sub>2</sub> {{=}} 1}} and {{math\|''g''<sub>3</sub> {{=}} 0}}. In the lemniscatic case, the minimal half period {{math\|''ω''<sub>1</sub>}} is real and equal to :<math>\frac{\Gamma^2\left(\frac{1}{4}\right)}{4\sqrt{\pi}}</math> where {{math\|Γ}} is the [[gamma function]]. The second smallest half period is pure imaginary and equal to {{math\|''iω''<sub>1</sub>}}. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s. The [[mathematical constant\|constant]]s {{math\|''e''<sub>1</sub>}}, {{math\|''e''<sub>2</sub>}}, and {{math\|''e''<sub>3</sub>}} are given by :<math>e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12. </math> The case {{math\|''g''<sub>2</sub> {{=}} ''a''}}, {{math\|''g''<sub>3</sub> {{=}} 0}} may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: {{math\|''a'' > 0}} and {{math\|''a'' < 0}}. The period [[parallelogram]] is either a [[square]] or a [[rhombus]]. =={{anchor\|sl\|cl}}Lemniscate sine and cosine functions== The '''lemniscate sine''' (Latin: ''sinus lemniscatus'') and '''lemniscate cosine''' (Latin: ''cosinus lemniscatus'') functions '''{{math\|sinlemn}}''' aka '''{{math\|sl}}''' and '''{{math\|coslemn}}''' aka '''{{math\|cl}}''' are analogues of the usual [[sine]] and [[cosine]] functions, with a circle replaced by a [[lemniscate]]. They are defined by :<math>\operatorname{sl}(r)=s</math> where :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}</math> and :<math>\operatorname{cl}(r)=c</math> where :<math> r=\int_c^1\frac{dt}{\sqrt{1-t^4}}.</math> They are doubly periodic (or elliptic) functions in the complex plane, with periods {{math\|2{{pi}}''G''}} and {{math\|2{{pi}}''iG''}}, where [[Gauss's constant]] {{math\|''G''}} is given by :<math>G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots.</math> ===Arclength of lemniscate=== [[Image:Lemniscate of Bernoulli.svg\|thumb\|400px\|right\|A lemniscate of Bernoulli and its two foci]] The [[lemniscate of Bernoulli]] :<math>\left(x^2+y^2\right)^2=x^2-y^2</math> consists of the points such that the product of their distances from the two points {{math\|({{sfrac\|1\|{{sqrt\|2}}}}, 0)}}, {{math\|(−{{sfrac\|1\|{{sqrt\|2}}}}, 0)}} is the constant {{sfrac\|1\|2}}. The length {{math\|''r''}} of the arc from the origin to a point at distance {{math\|''s''}} from the origin is given by :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.</math> In other words, the sine lemniscatic function gives the distance from the origin as a function of the arc length from the origin. Similarly the cosine lemniscate function gives the distance from the origin as a function of the arc length from (1, 0). =={{anchor\|arcsl\|arccl}}Inverse functions== {{Empty section\|date=August 2017\|reason=Inverse functions are named arcsl and arccl and should be discussed here.}} ==See also== [[Gauss's constant]] [[Equianharmonic\|Equianharmonic case]] ==References== {{AS ref\|18\|658}} {{dlmf \|id=23.5.iii \|title=Lemniscate lattice \|first1=W. P. \|last1=Reinhardt \|first2=P. L. \|last2=Walker}} {{cite journal \|mr=0257326 \|author-last=Siegel \|author-first=C. L. \|title=Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory \|series=Interscience Tracts in Pure and Applied Mathematics \|volume=25 \|publisher=Wiley-Interscience A Division of John Wiley & Sons \|place=New York-London-Sydney \|date=1969 \|isbn=0-471-60844-0}} ==External links== {{springer\|title=Lemniscate functions\|id=p/l058120}} [[Category:Modular forms]] [[Category:Elliptic curves]] [[Category:Lemniscatic elliptic functions\| ]]
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LaTeX to CAS translator

Mathematica

Information

Tests

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Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

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Tests

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Numeric

Dependency Graph Information

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