Latest revision as of 12:55, 28 November 2023

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.

For legal inquiries, use the methods described at https://wikitech.wikimedia.org/.

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(- zt)(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[- zt](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.

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

↑ 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

[Abdou2020-1] 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

[1]

@@ Line 1: / Line 1: @@
-This page presents the results of the publication: ''Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems''.
+This wiki provides an overview of LaCASt, our LaTeX to Computer Algebra Systems (CAS) Translator, and its translations of the [https://dlmf.nist.gov/ 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.
+For legal inquiries, use the methods described at https://wikitech.wikimedia.org/.
 = Bug Reports =
@@ Line 10: / Line 14: @@
 = DLMF Translations and Results =
-In the following, we present the translations of the [https://dlmf.nist.gov/ DLMF] equations to the CAS [https://www.maplesoft.com/ Maple] and [https://www.wolfram.com/mathematica/ Mathematica].
+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 [https://dlmf.nist.gov/ 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 [https://dlmf.nist.gov/4.12.E1 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 <code>Skipped - no semantic math</code>). 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.
-{| class="wikitable sortable"
+<div style="-moz-column-count:2; column-count:2;">
-|-
+The most-right columns in the table show the numeric evaluation results. If an equation was not numerically verified, it shows <code>Failed [x/y]</code> where <math>x</math> is the number of failed calculations and <math>y</math> is the number of all test calculations for this test case. As you can see in the second entry below [https://dlmf.nist.gov/11.5.E2 11.5.E2], 15 test calculations failed out of 25, so 10 were successful. When you click on the little <code>Expand</code>, 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.
-! DLMF !! Formula !! Translations<br>Maple !! Translations<br>Mathematica !! Symbolic Evaluation<br>Maple !! Symbolic Evaluation<br>Mathematica !! Numeric Evaluation<br>Maple !! Numeric Evaluation<br>Mathematica
-|-
+Clicking on <code>expand</code> 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 <code>LHS - RHS</code>. Further explanations on why this test failed can be found in the paper.
-| [https://dlmf.nist.gov/ DLMF] || 6,623 || 4,114 (62.1%) || 4,713 (71.2%) || 1,084 (26.3%) || 1,235 (26.2%) || 698 (26.7%) || 784 (22.6%)
-|-
-|}
-By clicking on a chapter of the [https://dlmf.nist.gov/ DLMF] below, you will see a large table that looks like this:
+<div class="toccolours">Failed [15 / 25]
+<syntaxhighlight lang=mathematica>Result: Complex[0.9495382353861556, -0.46093572348323536]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}</syntaxhighlight>
+<syntaxhighlight lang=mathematica>Result: Complex[0.7706973036767981, -0.20650772012904173]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}</syntaxhighlight>... skip entries to safe data</div>
+</div>
+<br>
-<div style="width: 100%; height: 75vh; overflow: auto;">
+<div style="width: 100%; overflow: auto;">
 {| class="wikitable sortable" style="margin: 0;"
 |-
@@ Line 41: / Line 54: @@
 Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.7706973036767981, -0.20650772012904173]
 Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|}
+</div>
+As a summary, here is an explanation for every column of the result tables.
+<div style="-moz-column-count:2; column-count:2;">
+; 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 <ref name="Abdou2020">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: [https://doi.org/10.1007/978-3-030-53518-6\_25 10.1007/978-3-030-53518-6_25]</ref>
+; 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, [https://dlmf.nist.gov/11.5.E2 11.5.E2] only has the constraint <math>\realpart@@{z} > 0</math>. 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.
+</div>
+== Links to the data ==
+In the following, we present an overview of the translations of the [https://dlmf.nist.gov/ DLMF] equations to the CAS [https://www.maplesoft.com/ Maple] and [https://www.wolfram.com/mathematica/ Mathematica].
+{| class="wikitable sortable"
+|-
+! DLMF !! Formula !! Translations<br>Maple !! Translations<br>Mathematica !! Symbolic Evaluation<br>Maple !! Symbolic Evaluation<br>Mathematica !! Numeric Evaluation<br>Maple !! Numeric Evaluation<br>Mathematica
+|-
+| [https://dlmf.nist.gov/ 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%)
 |-
-|-
-| [https://dlmf.nist.gov/18.5.E8 18.5.E8] || [[Item:Q5516|<math>\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 81] || Successful [Tested: 81]
-|-
-| [https://dlmf.nist.gov/18.5.E8 18.5.E8] || [[Item:Q5516|<math>2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 81] || Successful [Tested: 81]
 |}
-</div>
-The result tables do not contain every equation with a label in the [https://dlmf.nist.gov/ DLMF] since quite a few equations were skipped (see explanations in the paper).
+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.
-== Translations and Evaluations of the Digital Library of Mathematical Functions ==
 <div style="-moz-column-count:2; column-count:2;">
 # [[Results of Algebraic and Analytic Methods|Algebraic and Analytic Methods]]
@@ Line 62: / Line 93: @@
 # [[Results of Incomplete Gamma and Related Functions|Incomplete Gamma and Related Functions]]
 # [[Results of Airy and Related Functions|Airy and Related Functions]]
-# [[Results of Bessel Functions I|Bessel Functions I]] & [[Results of Bessel Functions II|Bessel Functions II]]
+# [[Results of Bessel Functions|Bessel Functions]]
 # [[Results of Struve and Related Functions|Struve and Related Functions]]
 # [[Results of Parabolic Cylinder Functions|Parabolic Cylinder Functions]]
@@ Line 71: / Line 102: @@
 # [[Results of q-Hypergeometric and Related Functions|''q''-Hypergeometric and Related Functions]]
 # [[Results of Orthogonal Polynomials|Orthogonal Polynomials]]
-# [[Results of Elliptic Integrals I|Elliptic Integrals I]] & [[Results of Elliptic Integrals II|Elliptic Integrals II]]
+# [[Results of Elliptic Integrals|Elliptic Integrals]]
 # [[Results of Theta Functions|Theta Functions]]
 # [[Results of Multidimensional Theta Functions|Multidimensional Theta Functions]]
@@ Line 90: / Line 121: @@
 # [[Results of Integrals with Coalescing Saddles|Integrals with Coalescing Saddles]]
 </div>
+=== Examples ===
+To provide you a starting point to investigate the data above, we suggest [[15.4|Hypergeometric Function - 15.4 Special Cases]]. For example, [https://dlmf.nist.gov/15.4#E28 15.4.E28]
+:<math>\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}}}</math>
+was successfully translated to Maple and Mathematica and verified symbolically and numerically in both systems despite the complexity of such a test case.
+{| class="wikitable"
+|-
+! Maple !! Mathematica
+|-
+| <syntaxhighlight lang=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)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]]</syntaxhighlight>
+|}
+Another interesting section might be [[26.3|Combinatorial Analysis - 26.3 Lattice Paths: Binomial Coefficients]]. All expressions are relatively straight-forward. Nonetheless, [https://dlmf.nist.gov/26.3#E8 23.3.E8]
+:<math>\binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}</math>
+failed numerically. This test case failed because Mathematica simplified the right-hand side directly to <math>\frac{(1+n) {m \choose n+1}}{m-n}</math> which would result in a division by zero for <math>m = n</math>. Indeed, our result table show that the test evaluation failed for <math>m=n=1</math> and <math>m=n=2</math> (the last case <math>m=n=3</math> 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 ==
@@ Line 163: / Line 211: @@
 ||   7 || 14.9% ||  33 || [  5 /  28] ||   1
 | style="border-right: solid 2px #aaa;" |   5
-||   4 ||  6.8% ||  38 || [  6 /  32] || {| class="wikitable sortable"
+||   4 ||  6.8% ||  38 || [  6 /  32] ||   7 ||   9
-|-
-! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
-|-  7 ||   9
 |-
 ! scope="row" style="text-align: left; border-right: solid 1px #000"|  3. [[Results of Numerical Methods|NM]]
@@ Line 252: / Line 297: @@
 ||  30 || 27.3% ||  58 || [ 38 /  20] ||  14 ||   7
 |-
-! scope="row" style="text-align: left; border-right: solid 1px #000"| 10. BS [[Results of Bessel Functions I|I]] & [[Results of Bessel Functions II|II]]
+! scope="row" style="text-align: left; border-right: solid 1px #000"| 10. [[Results of Bessel Functions|BS]]
 | style="border-right: solid 1px #000;" | 653
 || 143 || 392
@@ Line 360: / Line 405: @@
 ||  45 || 24.3% ||  68 || [ 31 /  37] ||  52 ||  12
 |-
-! scope="row" style="text-align: left; border-right: solid 1px #000"| 19. EL [[Results of Elliptic Integrals I|I]] & [[Results of Elliptic Integrals II|II]]
+! scope="row" style="text-align: left; border-right: solid 1px #000"| 19. [[Results of Elliptic Integrals|EL]]
 | style="border-right: solid 1px #000;" | 516
 || 103 || 252
@@ Line 385: / Line 430: @@
 |-
 ! scope="row" style="text-align: left; border-right: solid 1px #000"| 21. [[Results of Multidimensional Theta Functions|MT]]
-|style="border-right: solid 1px #000;" | -
+|style="border-right: solid 1px #000;" | 32
-|| - || -
+|| 0 || 0
-| style="border-right: solid 1px #000;" | -
+| style="border-right: solid 1px #000;" | 0
 || - || -
 | style="border-right: solid 2px #aaa;" | -
@@ Line 553: / Line 598: @@
 |-
 ! scope="row" style="text-align: left; border-right: solid 1px #000"| 35. [[Results of Functions of Matrix Argument|FM]]
-|style="border-right: solid 1px #000;" | -
+|style="border-right: solid 1px #000;" | 46
-|| - || -
+|| 0 || 0
-| style="border-right: solid 1px #000;" | -
+| style="border-right: solid 1px #000;" | 0
 || - || -
 | style="border-right: solid 2px #aaa;" | -
@@ Line 576: / Line 621: @@
 ||   3 || 12.5% ||  13 || [  1 /  12] ||   1 ||   6
 |-
-! <math>\sum</math> || 6545 || 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
+! <math>\sum</math> || 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=

Verifying DLMF with Maple and Mathematica: Difference between revisions

Latest revision as of 12:55, 28 November 2023

Contents

Bug Reports

DLMF Translations and Results

How to read the data?

Links to the data

Examples

Translations and Evaluations Overview Table

References

Navigation menu

Verifying DLMF with Maple and Mathematica: Difference between revisions

Latest revision as of 12:55, 28 November 2023

Bug Reports

DLMF Translations and Results

How to read the data?

Links to the data

Examples

Translations and Evaluations Overview Table

References

Navigation menu

Search