This shortening feature of integration which will

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this shortening feature of integration, which will prove wrong on a FWD test set. Similar problems will happen on a FWD trained model with a BWD test set. Since IBP keeps average expression lengths unchanged, BWD and FWD -trained models will generalize better to IBP test sets (and be more accurate on FWD -trained models, since their input length distributions are closer). Figure 2: Distribution of input and output lengths for different integration datasets. The FWD generator produces short problems with long solutions. Conversely, the BWD generator creates long problems, with short solutions. The IBP approach stands in the middle, and generates short problems with short solutions. This suggests that what looks at first glance like a generalization problem (bad accuracy of BWD - trained models on FWD generated sets, and the converse) is in fact a consequence of data generation. BWD and FWD methods generate training sets with specific properties, that our model will learn. But this can be addressed by adding IBP or FWD data to the BWD dataset, as shown in the two last lines of Table 6. In practice, a better approach could be implemented with self-supervised learning, where new training examples are generated by the model itself. 23
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Functions and their primitives generated with the forward approach ( FWD ) cos -1 ( x ) x cos -1 ( x ) - p 1 - x 2 x (2 x + cos (2 x )) 2 x 3 3 + x sin (2 x ) 2 + cos (2 x ) 4 x ( x + 4) x + 2 x 2 2 + 2 x - 4 log ( x + 2) cos (2 x ) sin ( x ) log (cos ( x ) - 1) 2 - log (cos ( x ) + 1) 2 + 2 cos ( x ) 3 x 2 sinh -1 (2 x ) x 3 sinh -1 (2 x ) - x 2 4 x 2 + 1 6 + 4 x 2 + 1 12 x 3 log ( x 2 ) 4 x 4 log ( x 2 ) 4 4 - x 4 log ( x 2 ) 3 2 + 3 x 4 log ( x 2 ) 2 4 - 3 x 4 log ( x 2 ) 4 + 3 x 4 8 Functions and their primitives generated with the backward approach ( BWD ) cos ( x ) + tan 2 ( x ) + 2 x + sin ( x ) + tan ( x ) 1 x 2 x - 1 x + 1 x - 1 x + 1 x 2 x cos 2 ( x ) + tan ( x ) tan ( x ) x tan 2 ( x ) x tan e x x + ( x - 1) e x cos 2 ( e x x ) x x tan e x x 1 + 1 log (log ( x )) - 1 log ( x ) log (log ( x )) 2 x + x log (log ( x )) - 2 x 2 sin ( x 2 ) tan ( x ) + x ( tan 2 ( x ) + 1 ) cos ( x 2 ) + cos ( x 2 ) tan ( x ) x cos ( x 2 ) tan ( x ) Functions and their primitives generated with the integration by parts approach ( IBP ) x ( x + log ( x )) x 2 (4 x + 6 log ( x ) - 3) 12 x ( x + 3) 2 - x + ( x + 3) log ( x + 3) x + 3 x + 2 cos 2 ( x ) x + 2 tan ( x ) + log (cos ( x )) x (2 x + 5) (3 x + 2 log ( x ) + 1) x 2 ( 27 x 2 + 24 x log ( x ) + 94 x + 90 log ( x ) ) 18 x - 2 x sin 2 ( x ) + 1 tan ( x ) log ( x ) sin ( x ) x log ( x ) + tan ( x ) sin ( x ) tan ( x ) x 3 sinh ( x ) x 3 cosh ( x ) - 3 x 2 sinh ( x ) + 6 x cosh ( x ) - 6 sinh ( x ) Table 9: Examples of functions with their integrals, generated by our FWD , BWD and IBP approaches. We observe that the FWD and IBP approaches tend to generate short functions, with long integrals, while the BWD approach generates short functions with long derivatives. 24
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