Verifying DLMF with Maple and Mathematica

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This page presents the results of the publication: Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems.

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

In the following, we present 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 6,623 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 DLMF below, you will see a large table that looks like this:

DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
Link to DLMF Formula Translation to Maple Translation to Mathematica Symbolic Result of Maple Symbolic Result of Mathematica Numeric Result of Maple Numeric Result of Mathematica

The result tables do not contain every equation with a label in the DLMF since quite a few equations were skipped (see explanations in the paper).

Translations and Evaluations of the Digital Library of Mathematical Functions

  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 I & Bessel Functions II
  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 I & Elliptic Integrals II
  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

Translations and Evaluations Overview Table

Meaning
2C Chapter Code
S Successful
% Percentage
F Fail
P/T Partially / Totally Failed
A Aborted
E Errors
Symbolic Numeric
Translations Maple Mathematica Maple Mathematica
2C Total Maple Math S % F S % F S % F [P/T] A E S % F [P/T] A E
1. AL 227 102 103 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 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 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 569 494 564 270 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 144 130 139 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 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 149 101 120 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 204 160 163 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 235 180 179 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 I & II 653 392 486 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 124 121 112 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 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 260 252 254 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 238 230 229 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 206 198 197 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 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 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 468 235 288 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 I & II 516 252 416 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 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 0 0 - - - - - - - - - - - - - - - - - - -
22. JA 264 232 238 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 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 175 117 148 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 154 124 120 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 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 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 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 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 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 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 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 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 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 0 0 - - - - - - - - - - - - - - - - - - -
36. IC 106 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
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