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 r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.}

${\displaystyle r=\int _{0}^{s}{\frac {dt}{\sqrt {1-t^{4}}}}.}$

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

Semantic latex: r = \int_0^s \frac{\diff{t}}{\sqrt{1-t^4}}

Confidence: 0

Mathematica

Translation: r == Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None]

Information

Sub Equations

r = Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None]

Free variables

r

s

Tests

Symbolic

Test expression: (r)-(Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None])

ERROR:

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

Numeric

SymPy

Translation: r == integrate((1)/(sqrt(1 - (t)**(4))), (t, 0, s))

Information

Sub Equations

r = integrate((1)/(sqrt(1 - (t)**(4))), (t, 0, s))

Free variables

r

s

Tests

Symbolic

Numeric

Maple

Translation: r = int((1)/(sqrt(1 - (t)^(4))), t = 0..s)

Information

Sub Equations

r = int((1)/(sqrt(1 - (t)^(4))), t = 0..s)

Free variables

r

s

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$r$

$s$

Description

length

arc from the origin

point at distance

origin

Complete translation information:

{
  "id" : "FORMULA_0ba4b0743a00ca9874293a3391755220",
  "formula" : "r=\\int_0^s\\frac{dt}{\\sqrt{1-t^4}}",
  "semanticFormula" : "r = \\int_0^s \\frac{\\diff{t}}{\\sqrt{1-t^4}}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "r == Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "r = Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None]" ],
        "freeVariables" : [ "r", "s" ]
      },
      "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" : "r",
          "rhs" : "Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None]",
          "testExpression" : "(r)-(Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, s}, GenerateConditions->None])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "r == integrate((1)/(sqrt(1 - (t)**(4))), (t, 0, s))",
      "translationInformation" : {
        "subEquations" : [ "r = integrate((1)/(sqrt(1 - (t)**(4))), (t, 0, s))" ],
        "freeVariables" : [ "r", "s" ]
      }
    },
    "Maple" : {
      "translation" : "r = int((1)/(sqrt(1 - (t)^(4))), t = 0..s)",
      "translationInformation" : {
        "subEquations" : [ "r = int((1)/(sqrt(1 - (t)^(4))), t = 0..s)" ],
        "freeVariables" : [ "r", "s" ]
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 1,
    "word" : 6
  }, {
    "section" : 2,
    "sentence" : 1,
    "word" : 21
  } ],
  "includes" : [ "r", "s" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "length",
    "score" : 0.44236504421594536
  }, {
    "definition" : "arc from the origin",
    "score" : 0.4025311038849381
  }, {
    "definition" : "point at distance",
    "score" : 0.30655166268737566
  }, {
    "definition" : "origin",
    "score" : 0.22301573207376618
  } ]
}

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

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Tests

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Numeric

SymPy

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Tests

Symbolic

Numeric

Maple

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Tests

Symbolic

Numeric

Dependency Graph Information

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