fix(polys): fix apart(full=True) with floats

[oscarbenjamin]

Don't assume that the factors divide the original polynomial exactly because if the domain is inexact then they might not. Instead compute the quotient and discard the remainder.

Fixes gh-26648

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Brief description of what is fixed or changed

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Release Notes

• polys
  • A bug in apart(full=True) was fixed. Previously incorrect results might be returned if the expression contained floats.

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• polys
  • A bug in apart(full=True) was fixed. Previously incorrect results might be returned if the expression contained floats. (#26649 by @oscarbenjamin)

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Don't assume that the factors divide the original polynomial exactly because if the domain is inexact then they might not. Instead compute the quotient and discard the remainder.

Fixes gh-26648

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* polys
  * A bug in `apart(full=True)` was fixed. Previously incorrect results might be returned if the expression contained floats.
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Update

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[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

| Change   | Before [a36a8b23] <sympy-1.12.1^0>   | After [71005aa0]    |   Ratio | Benchmark (Parameter)                                                |
|----------|--------------------------------------|---------------------|---------|----------------------------------------------------------------------|
| -        | 67.8±1ms                             | 43.7±0.1ms          |    0.64 | integrate.TimeIntegrationRisch02.time_doit(10)                       |
| -        | 67.0±0.3ms                           | 42.6±0.2ms          |    0.64 | integrate.TimeIntegrationRisch02.time_doit_risch(10)                 |
| +        | 18.6±0.4μs                           | 30.4±0.3μs          |    1.63 | integrate.TimeIntegrationRisch03.time_doit(1)                        |
| -        | 5.42±0.01ms                          | 2.90±0.02ms         |    0.53 | logic.LogicSuite.time_load_file                                      |
| -        | 73.4±0.6ms                           | 28.5±0.1ms          |    0.39 | polys.TimeGCD_GaussInt.time_op(1, 'dense')                           |
| -        | 25.9±0.06ms                          | 16.9±0.06ms         |    0.65 | polys.TimeGCD_GaussInt.time_op(1, 'expr')                            |
| -        | 74.0±0.2ms                           | 28.9±0.1ms          |    0.39 | polys.TimeGCD_GaussInt.time_op(1, 'sparse')                          |
| -        | 256±2ms                              | 124±0.2ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'dense')                           |
| -        | 256±2ms                              | 125±1ms             |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'sparse')                          |
| -        | 656±4ms                              | 375±2ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'dense')                           |
| -        | 658±2ms                              | 372±1ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'sparse')                          |
| -        | 499±3μs                              | 290±2μs             |    0.58 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(1, 'dense')            |
| -        | 1.78±0.02ms                          | 1.04±0.01ms         |    0.58 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(2, 'dense')            |
| -        | 5.85±0.02ms                          | 3.08±0.01ms         |    0.53 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(3, 'dense')            |
| -        | 446±2μs                              | 229±1μs             |    0.51 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(1, 'dense')               |
| -        | 1.47±0.01ms                          | 683±10μs            |    0.46 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(2, 'dense')               |
| -        | 4.93±0.02ms                          | 1.67±0.01ms         |    0.34 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(3, 'dense')               |
| -        | 375±1μs                              | 206±1μs             |    0.55 | polys.TimeGCD_SparseGCDHighDegree.time_op(1, 'dense')                |
| -        | 2.43±0.02ms                          | 1.24±0ms            |    0.51 | polys.TimeGCD_SparseGCDHighDegree.time_op(3, 'dense')                |
| -        | 10.1±0.1ms                           | 4.33±0.03ms         |    0.43 | polys.TimeGCD_SparseGCDHighDegree.time_op(5, 'dense')                |
| -        | 361±4μs                              | 169±0.9μs           |    0.47 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(1, 'dense')            |
| -        | 2.51±0.07ms                          | 893±8μs             |    0.36 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(3, 'dense')            |
| -        | 9.73±0.1ms                           | 2.64±0.03ms         |    0.27 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(5, 'dense')            |
| -        | 1.04±0.01ms                          | 429±4μs             |    0.41 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'dense')           |
| -        | 1.74±0ms                             | 506±0.5μs           |    0.29 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'sparse')          |
| -        | 5.96±0.03ms                          | 1.78±0.02ms         |    0.3  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'dense')           |
| -        | 8.56±0.04ms                          | 1.48±0.01ms         |    0.17 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'sparse')          |
| -        | 289±2μs                              | 64.7±0.3μs          |    0.22 | polys.TimePREM_QuadraticNonMonicGCD.time_op(1, 'sparse')             |
| -        | 3.45±0.03ms                          | 390±4μs             |    0.11 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'dense')              |
| -        | 4.02±0.02ms                          | 282±2μs             |    0.07 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'sparse')             |
| -        | 7.08±0.09ms                          | 1.26±0.01ms         |    0.18 | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'dense')              |
| -        | 8.78±0.04ms                          | 831±4μs             |    0.09 | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'sparse')             |
| -        | 5.11±0.04ms                          | 2.98±0.01ms         |    0.58 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(2, 'sparse') |
| -        | 12.1±0.05ms                          | 6.67±0.02ms         |    0.55 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'dense')  |
| -        | 22.5±0.05ms                          | 9.01±0.02ms         |    0.4  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'sparse') |
| -        | 5.22±0.01ms                          | 868±3μs             |    0.17 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(1, 'sparse')    |
| -        | 12.7±0.04ms                          | 6.97±0.02ms         |    0.55 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(2, 'sparse')    |
| -        | 103±0.6ms                            | 25.7±0.1ms          |    0.25 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'dense')     |
| -        | 166±0.4ms                            | 53.3±0.2ms          |    0.32 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'sparse')    |
| -        | 177±0.6μs                            | 113±2μs             |    0.64 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'dense')      |
| -        | 360±2μs                              | 218±2μs             |    0.6  | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'sparse')     |
| -        | 4.30±0.02ms                          | 859±6μs             |    0.2  | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'dense')      |
| -        | 5.27±0.03ms                          | 382±2μs             |    0.07 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'sparse')     |
| -        | 20.0±0.4ms                           | 2.80±0.01ms         |    0.14 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'dense')      |
| -        | 22.7±0.1ms                           | 626±5μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'sparse')     |
| -        | 482±1μs                              | 135±0.6μs           |    0.28 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(1, 'sparse') |
| -        | 4.83±0.01ms                          | 617±3μs             |    0.13 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'dense')  |
| -        | 5.31±0.04ms                          | 141±1μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'sparse') |
| -        | 13.3±0.1ms                           | 1.30±0.01ms         |    0.1  | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'dense')  |
| -        | 14.1±2ms                             | 144±1μs             |    0.01 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'sparse') |
| -        | 136±0.7μs                            | 74.9±0.6μs          |    0.55 | solve.TimeMatrixOperations.time_rref(3, 0)                           |
| -        | 256±0.5μs                            | 89.7±0.9μs          |    0.35 | solve.TimeMatrixOperations.time_rref(4, 0)                           |
| -        | 24.5±0.5ms                           | 10.2±0.02ms         |    0.41 | solve.TimeSolveLinSys189x49.time_solve_lin_sys                       |
| -        | 29.2±0.2ms                           | 15.4±0.1ms          |    0.53 | solve.TimeSparseSystem.time_linsolve_Aaug(20)                        |
| -        | 56.6±0.2ms                           | 24.7±0.08ms         |    0.44 | solve.TimeSparseSystem.time_linsolve_Aaug(30)                        |
| -        | 29.1±0.1ms                           | 15.1±0.02ms         |    0.52 | solve.TimeSparseSystem.time_linsolve_Ab(20)                          |
| -        | 56.7±0.2ms                           | 24.5±0.08ms         |    0.43 | solve.TimeSparseSystem.time_linsolve_Ab(30)                          |

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[asmeurer]

This looks fine. I bet there's a lot of code that does this sort of thing.

[asmeurer]

We should make all_close smart enough that this sort of thing isn't needed. I guess that might not be straightforward to do.

[oscarbenjamin]

It is not hard to do but it would necessarily be horribly inefficient.

[asmeurer]

Yeah, I meant to do efficiently. Your implementation should work. Hopefully in most cases you either get a fast exit or the expressions are sorted closely enough to each other that it isn't too quadratic.

[oscarbenjamin]

Note that "too quadratic" means "doubly exponential" for deep expression trees.