Verifying DLMF with Maple and Mathematica

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This wiki provides an overview of LaCASt, our LaTeX to Computer Algebra Systems (CAS) Translator, and its translations of the Digital Library of Mathematical Functions (DLMF). Specifically, we provide access to the translations for Maple and Mathematica. Additionally, each translation was verified with symbolic and numeric evaluation techniques. In the following, we present the results of our translations and evaluations.

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Bug Reports

You can find a PDF with commands that illustrate the encountered errors in Mathematica here: File:Mathematica Bugs Overview.pdf

We provide the same file in the Wolfram system notebook format (NB) here: File:Mathematica Bugs Notebook File.nb

DLMF Translations and Results

This section explains the data you can access. Below, you will find a large table that shows the overall results of translations, symbolic and numeric verifications, as well as a number of errors and aborted calculations for each DLMF chapter. Before you move on to investigating the results, please have a look at how to read the data first.

How to read the data?

The data on this website can be considered as an extension of the DLMF but not a copy. Some labeled equations in the DLMF do not appear in this data because they were skipped or caused translation errors in both systems (Maple and Mathematica). In addition, the result tables on this website do not provide context! For checking the context, every equation in our dataset is linked directly to the equation in the DLMF. You can access the original DLMF equation by clicking on the label in the most-left column in the result tables.

You can access the results by clicking on a particular chapter below. This will load a page showing links to each section. Red links indicate that this section is empty, i.e., no results are available for that section. Clicking on a blue-inked link will load the result table for that section. Below you see an example of such a table with just two equations from different chapters. As you can see, the first entry 4.12.E1 is grayed out. This happens when the entry was skipped due to missing semantic macros in the source of the test case (flagged as Skipped - no semantic math). You can see the actual semantic LaTeX source of every test case below the equation. Since most of these cases are not translated properly by LCT, we skipped them directly, i.e., they were not further processed for symbolic or numeric evaluations. In fact, those cases have not even been translated into our pipeline. However, we added the translations just for our result tables because they may provide further interesting insights into the translation process.

The most-right columns in the table show the numeric evaluation results. If an equation was not numerically verified, it shows Failed [x/y] where Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x} is the number of failed calculations and Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y} is the number of all test calculations for this test case. As you can see in the second entry below 11.5.E2, 15 test calculations failed out of 25, so 10 were successful. When you click on the little Expand, we show the first couple of results for the failed test cases. Since the tables are already very large, we cannot show all failed test calculations.

Clicking on expand for the second entry in our example table below, we see the test result and the test values that produced this result. As explained in the paper, the calculation was simply performed on the difference of the left- and right-hand sides of the equation LHS - RHS. Further explanations on why this test failed can be found in the paper.

Failed [15 / 25] 
Result: Complex[0.9495382353861556, -0.46093572348323536]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}

Result: Complex[0.7706973036767981, -0.20650772012904173]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}
... skip entries to safe data


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi(x+1) = e^{\phi(x)}}
\phi(x+1) = e^{\phi(x)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < x, x < \infty} 
phi(x + 1) = exp(phi(x))
 
\[Phi][x + 1] == Exp[\[Phi][x]]
 Skipped - no semantic math Skipped - no semantic math - -
11.5.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}}
\StruveK{\nu}@{z} = \frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0, \realpart@@{(\nu+\tfrac{1}{2})} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(n+\nu+\tfrac{3}{2})} > 0} 
StruveH(nu, z) - BesselY(nu, z) = (2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity)

StruveH[\[Nu], z] - BesselY[\[Nu], z] == Divide[2*(Divide[1,2]*z)^\[Nu],Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}, GenerateConditions->None]
Successful Successful -
Failed [15 / 25]
Result: Complex[0.9495382353861556, -0.46093572348323536]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}


Result: Complex[0.7706973036767981, -0.20650772012904173]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}

... skip entries to safe data

As a summary, here is an explanation for every column of the result tables.

DLMF
Contains the equation label as a link to the actual equation in the DLMF. Click on the label if you need to check the context of a formula in the DLMF.
Formula
Contains the rendered formula and the actual semantic LaTeX source (sometimes also referred to as content-tex). More information about the semantic (or content) LaTeX can be found in [1]
Constraints
This shows the constraints that LCT has generated by analyzing the definitions of the functions that are included in the equation. This also includes the constraints that are directly attached to a test case. For example, 11.5.E2 only has the constraint Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0} . LCT followed the definitions of the involved functions and identified several more constraints for this test case. A detailed explanation of this process was given in the paper. Note, however, that this list of constraints does not contain the manually defined global constraints that we applied for all test cases (as shown in Table 3 in the paper). This way, the result tables only show what LCT generated and applied for the numeric evaluations.
Maple
The translation to Maple
Mathematica
The translation to Mathematica
Symbolic (Maple / Mathematica)
The result of the symbolic evaluation. If this was skipped, no calculations have been performed on the test case.
Numeric (Maple / Mathematica)
The result of the numeric evaluation as explained above and in the paper.

Links to the data

In the following, we present an overview of the translations of the DLMF equations to the CAS Maple and Mathematica.

DLMF Formula Translations
Maple		 Translations
Mathematica		 Symbolic Evaluation
Maple		 Symbolic Evaluation
Mathematica		 Numeric Evaluation
Maple		 Numeric Evaluation
Mathematica
DLMF 6,545 4,114 (62.9%) 4,713 (72.0%) 1,084 (26.3%) 1,235 (26.2%) 698 (26.7%) 784 (22.6%)

The following links provide access to all results for each DLMF chapter. This will show the sections in each chapter. Please notice that many sections in a chapter might be empty (indicated by a red link color instead of a blue link). This happens when all formulae in a section have been skipped or the case analyzer was not able to parse the expression properly. In some cases, the section simply did not contain any formula.

  1. Algebraic and Analytic Methods
  2. Asymptotic Approximations
  3. Numerical Methods
  4. Elementary Functions
  5. Gamma Function
  6. Exponential, Logarithmic, Sine, and Cosine Integrals
  7. Error Functions, Dawson’s and Fresnel Integrals
  8. Incomplete Gamma and Related Functions
  9. Airy and Related Functions
  10. Bessel Functions
  11. Struve and Related Functions
  12. Parabolic Cylinder Functions
  13. Confluent Hypergeometric Functions
  14. Legendre and Related Functions
  15. Hypergeometric Function
  16. Generalized Hypergeometric Functions and Meijer G-Function
  17. q-Hypergeometric and Related Functions
  18. Orthogonal Polynomials
  19. Elliptic Integrals
  20. Theta Functions
  21. Multidimensional Theta Functions
  22. Jacobian Elliptic Functions
  23. Weierstrass Elliptic and Modular Functions
  24. Bernoulli and Euler Polynomials
  25. Zeta and Related Functions
  26. Combinatorial Analysis
  27. Functions of Number Theory
  28. Mathieu Functions and Hill’s Equation
  29. Lamé Functions
  30. Spheroidal Wave Functions
  31. Heun Functions
  32. Painlevé Transcendents
  33. Coulomb Functions
  34. 3j,6j,9j Symbols
  35. Functions of Matrix Argument
  36. Integrals with Coalescing Saddles

Examples

To provide you a starting point to investigate the data above, we suggest Hypergeometric Function - 15.4 Special Cases. For example, 15.4.E28

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \hyperF@{a}{b}{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}{\tfrac{1}{2}} = \sqrt{\pi}\frac{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}}

was successfully translated to Maple and Mathematica and verified symbolically and numerically in both systems despite the complexity of such a test case.

Maple Mathematica 
hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b +(1)/(2)], (1)/(2)) = sqrt(Pi)*(GAMMA((1)/(2)*a +(1)/(2)*b +(1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))

Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b +Divide[1,2], Divide[1,2]] == Sqrt[Pi]*Divide[Gamma[Divide[1,2]*a +Divide[1,2]*b +Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]

Another interesting section might be Combinatorial Analysis - 26.3 Lattice Paths: Binomial Coefficients. All expressions are relatively straight-forward. Nonetheless, 23.3.E8

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}}

failed numerically. This test case failed because Mathematica simplified the right-hand side directly to Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{(1+n) {m \choose n+1}}{m-n}} which would result in a division by zero for Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m = n} . Indeed, our result table show that the test evaluation failed for Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m=n=1} and Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m=n=2} (the last case Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m=n=3} was omitted because we only show the first two failed calculations to reduce the size of the result tables). This simple example shows the challenge of interpreting the data and working with computer algebra systems.

Translations and Evaluations Overview Table

Meaning
2C Chapter Code
S Successful
% Percentage
F Fail
P/T Partially / Totally Failed
A Aborted
E Errors
Base The baseline performance of the translator
Maple The CAS Maple 2020
Mathematica The CAS Mathematica
Symbolic Numeric
Formulae Translations Maple Mathematica Maple Mathematica
2C Total Base Maple Math S % F S % F S % F [P/T] A E S % F [P/T] A E
1. AL 227 60 102 103 36 35.3% 60 34 33.0% 69 14 23.3% 35 [ 12 / 23] 7 4 14 20.3% 40 [ 9 / 31] 11 4
2. AS 136 33 65 65 6 9.2% 47 6 9.2% 59 7 14.9% 33 [ 5 / 28] 1 5 4 6.8% 38 [ 6 / 32] 7 9
3. NM 53 36 40 40 6 15.0% 31 5 12.5% 35 1 3.2% 27 [ 9 / 18] 0 2 0 0.0% 29 [ 8 / 21] 6 0
4. EF 569 353 494 564 270 54.7% 221 304 53.9% 260 88 39.8% 126 [ 64 / 62] 0 6 110 42.3% 146 [ 55 / 91] 2 0
5. GA 144 38 130 139 41 31.5% 76 65 46.8% 74 39 51.3% 25 [ 12 / 13] 4 8 30 40.5% 20 [ 9 / 11] 13 9
6. EX 107 21 56 77 13 23.2% 43 18 23.4% 59 10 23.2% 31 [ 13 / 18] 0 2 23 39.0% 32 [ 6 / 26] 4 0
7. ER 149 35 101 120 52 51.5% 47 45 37.5% 75 21 44.7% 23 [ 10 / 13] 2 1 21 28.0% 43 [ 13 / 30] 9 1
8. IG 204 84 160 163 51 31.9% 102 65 39.9% 98 27 26.5% 61 [ 20 / 41] 9 5 22 22.4% 44 [ 19 / 25] 16 15
9. AI 235 36 180 179 54 30.0% 124 69 38.5% 110 34 27.4% 75 [ 41 / 34] 4 8 30 27.3% 58 [ 38 / 20] 14 7
10. BS 653 143 392 486 80 20.4% 209 115 23.7% 371 86 41.1% 59 [ 41 / 18] 52 12 90 24.2% 151 [ 57 / 94] 92 18
11. ST 124 48 121 112 39 32.2% 73 36 32.1% 76 25 34.2% 40 [ 14 / 26] 3 5 21 27.6% 33 [ 8 / 25] 10 11
12. PC 106 33 79 90 25 31.6% 50 18 20.0% 72 15 30.0% 24 [ 15 / 9] 11 0 13 18.0% 43 [ 15 / 28] 12 3
13. CH 260 126 252 254 75 29.8% 143 69 27.2% 185 14 9.8% 90 [ 55 / 35] 37 2 23 12.4% 95 [ 59 / 36] 45 21
14. LE 238 166 230 229 30 13.0% 163 30 13.1% 199 40 24.5% 93 [ 57 / 36] 18 12 59 29.6% 92 [ 54 / 38] 41 5
15. HY 206 148 198 197 46 23.2% 115 53 26.9% 144 17 14.8% 52 [ 34 / 18] 23 23 23 16.0% 77 [ 52 / 25] 29 6
16. GH 53 20 23 25 3 13.0% 16 2 8.0% 23 1 6.2% 9 [ 8 / 1] 6 0 1 4.3% 10 [ 7 / 3] 9 2
17. QH 175 1 53 124 23 43.4% 24 6 4.8% 118 0 0.0% 0 [ 0 / 0] 1 23 13 11.0% 57 [ 52 / 5] 39 5
18. OP 468 132 235 288 65 27.6% 148 101 35.1% 185 67 45.3% 50 [ 32 / 18] 14 17 45 24.3% 68 [ 31 / 37] 52 12
19. EL 516 103 252 416 39 15.5% 192 51 12.2% 365 18 9.4% 123 [ 44 / 79] 34 17 18 4.9% 264 [ 49 / 215] 61 15
20. TH 128 52 98 98 10 10.2% 68 1 1.0% 97 0 0.0% 32 [ 17 / 15] 20 16 33 34.0% 40 [ 25 / 15] 24 0
21. MT 32 0 0 0 - - - - - - - - - - - - - - - - - -
22. JA 264 115 232 238 46 19.8% 176 30 12.6% 206 20 11.4% 116 [ 25 / 91] 36 4 22 10.7% 131 [ 39 / 92] 51 0
23. WE 164 7 19 34 1 5.3% 16 4 11.8% 30 0 0.0% 14 [ 2 / 12] 1 1 2 6.7% 23 [ 9 / 14] 2 3
24. BP 175 31 117 148 15 12.8% 101 23 15.5% 125 67 66.3% 32 [ 19 / 13] 1 1 78 62.4% 33 [ 22 / 11] 14 0
25. ZE 154 28 124 120 19 15.3% 90 48 40.0% 72 43 47.8% 29 [ 18 / 11] 10 8 22 30.5% 22 [ 6 / 16] 22 3
26. CM 136 31 78 87 20 25.6% 50 19 21.8% 68 30 60.0% 11 [ 8 / 3] 2 7 44 64.7% 18 [ 10 / 8] 5 1
27. NT 79 5 26 15 3 11.5% 17 6 40.0% 9 2 11.8% 6 [ 3 / 3] 0 8 3 33.3% 6 [ 3 / 3] 0 0
28. MA 267 52 97 110 7 7.2% 80 7 6.4% 103 6 7.5% 32 [ 12 / 20] 26 15 3 2.9% 48 [ 13 / 35] 33 17
29. LA 111 11 23 22 0 0.0% 21 0 0.0% 22 0 0.0% 19 [ 2 / 17] 0 2 0 0.0% 21 [ 1 / 20] 0 1
30. SW 71 14 19 26 0 0.0% 18 0 0.0% 26 0 0.0% 18 [ 5 / 13] 0 0 0 0.0% 19 [ 2 / 17] 5 1
31. HE 35 29 22 15 5 22.7% 13 2 13.3% 13 2 15.4% 10 [ 0 / 10] 0 1 0 0.0% 8 [ 0 / 8] 5 0
32. PT 67 43 57 57 3 5.3% 51 3 5.3% 54 1 2.0% 44 [ 7 / 37] 4 2 0 0.0% 41 [ 2 / 39] 8 5
33. CW 108 21 14 11 1 7.1% 13 0 0.0% 11 0 0.0% 5 [ 2 / 3] 0 8 0 0.0% 11 [ 2 / 9] 0 0
34. TJ 57 0 1 37 0 0.0% 1 0 0.0% 37 0 0.0% 1 [ 0 / 1] 0 0 14 37.8% 10 [ 5 / 5] 13 0
35. FM 46 0 0 0 - - - - - - - - - - - - - - - - - -
36. IC 106 12 24 24 0 0.0% 19 0 0.0% 24 3 15.8% 12 [ 1 / 11] 3 1 3 12.5% 13 [ 1 / 12] 1 6
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum} 6623 2067 4114 4713 1084 26.3% 2618 1235 26.2% 3474 698 26.7% 1357 [607 / 750] 329 226 784 22.6% 1784 [687 / 1097] 655 180

References

  1. ↑ Abdou Youssef and Bruce R. Miller, "A Contextual and Labeled Math-Dataset Derived from NIST's DLMF." In: Intelligent Computer Mathematics (CICM) 2020: 324-330. DOI: 10.1007/978-3-030-53518-6_25
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