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Simplify does not cancel a factor when a numerator's term holds a power of it #2301
Description
On master (c633c41), Simplify() cancels a factor that every term of a numerator shares with the denominator, but not when a term holds a power of it:
Simplify |
Should be | |
|---|---|---|
(x * (y + 1) + (y + 1)) / (y + 1)^2 |
(x + 1) / (y + 1) |
the same |
(x * (y + 1) + (y + 1)^2) / (y + 1)^4 |
(x * (y + 1) + (y + 1) ^ 2) / (y + 1) ^ 4 |
(x + y + 1) / (y + 1) ^ 3 |
(x * (x^2 + 1) + (x^2 + 1)^2) / (x^2 + 1)^4 |
left as it is | (x ^ 2 + x + 1) / (x ^ 2 + 1) ^ 3 |
(a * (x^2 + 1) + b * (x^2 + 1)^2) / (x^2 + 1)^3 |
(a + b * x ^ 2 + b) / (x ^ 4 + 2 * x ^ 2 + 1) |
right, but expanded |
The values are right; the answer is not as far as it goes.
It is what makes a Taylor expansion of e^x / (1 + x^2) slow after #2291. The quotient rule squares the denominator at each step, and Simplify does not take back the factor the new numerator shares with it. So the simplified derivatives go e ^ x * (1 + x ^ 2 - 2 * x) / (1 + x ^ 2) ^ 2, then (… * (x ^ 2 + 1) * … + e ^ x * (x ^ 2 + 1) ^ 2 * …) / (x ^ 2 + 1) ^ 4, then a 15-term numerator over (x ^ 2 + 1) ^ 8. That is 21, 49 and 149 nodes, then 629 at the fifth and 5,117 at the seventh, and 9 terms take 8 s.
Part of #2133.
Activity
A Taylor expansion simplifies each derivative before taking the next (#…