Improve sympy import time by eliminating redundant ordering of Dispatcher

[eendebakpt]

Brief description of what is fixed or changed

During the sympy import several Dispatchers are created and methods are registered. On each registration the full set of methods is ordered, which is costly.

This PR reduces the sorting by

1. Stopping the sorting before importing subpackages
2. After importing subpackges (which includes the registration of methods to dispatchers), sort all dispatchers

Existing functionality of the Dispatcher is used.

The import time is reduced from 269 to 258 ms.

Output of bin/test_import

Master:

eendebakpt@woelmuis:/mnt/data/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.2750202629999876, 0.27497193600015635, 0.2702798200000416, 0.26889375400014615, 0.2687653349998982, 0.2716369390000182, 0.2794790110001486, 0.26655530899984115, 0.2691565279999395, 0.2684384439999121, 0.27359299299996565, 0.26817051800003355, 0.26997147999986737, 0.26848315500001263, 0.2695972559999973, 0.2689053330000206, 0.266499002999808, 0.27336413400007586, 0.2695815440001752, 0.2726911500001279, 0.2669147869999051, 0.26745493099997475, 0.27697483599990846, 0.26868902599994726, 0.2712855639999816, 0.26795200600008684, 0.268541746999972, 0.26775163699994664, 0.27113610199990035, 0.2674677069999234, 0.26795610500016664, 0.26821584900017115, 0.26595949700003985, 0.2713385689999086, 0.2666728149999926, 0.2681342289999975, 0.2687668170001416, 0.2691814659999636, 0.26562634600009005, 0.26544889799993143, 0.2664543210000829, 0.2697006410000995, 0.2687029410001287, 0.26908488100002614, 0.2688644029999523, 0.2712997159999304, 0.27255410699990534, 0.2704326430000492, 0.270944481000015, 0.2717818040000566, 0.2686064910001278]
Number of tests: 50
The speed of "import sympy" is: 0.269579 +- 0.002746
eendebakpt@woelmuis:/mnt/data/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.27012507099993854, 0.2690900089999104, 0.2714201490000505, 0.266401506999955, 0.269292661999998, 0.2699106360000769, 0.2750912349999908, 0.2690173099999811, 0.2662025030001587, 0.27493229600008817, 0.2732122560000789, 0.26986700499992367, 0.2673109410000052, 0.2685452460000306, 0.2705744620000132, 0.27552572399986275, 0.26664292999998906, 0.2703416899998956, 0.27229860700003883, 0.2681302329999653, 0.26659810000001016, 0.2696212359999208, 0.2680760770001598, 0.2669595599998047, 0.2689367319999292, 0.2691656189999776, 0.2738177270000506, 0.270048332999977, 0.27325413200014737, 0.26922133700009, 0.2687775770000371, 0.2711154799999349, 0.2729384140000093, 0.2654830999999831, 0.270534656999871, 0.27074404199993296, 0.2697775599999659, 0.2736168829999315, 0.26733127000011336, 0.2783130220000203, 0.2690745869999773, 0.2671201389998714, 0.26748783299990464, 0.2705764470001668, 0.26714791799986415, 0.26750925000010284, 0.2674039340001855, 0.2647815189998255, 0.2660385930000757, 0.2756670290000329, 0.26935999700003777]
Number of tests: 50
The speed of "import sympy" is: 0.269806 +- 0.002957

PR

eendebakpt@woelmuis:~/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.25928017599994746, 0.25534695700002885, 0.25327333200016255, 0.2605424830001084, 0.25428364799995506, 0.25779407999993964, 0.25567224200017336, 0.255802980999988, 0.2575704589999077, 0.2656662080000842, 0.25502123999990545, 0.25856492699995215, 0.2610679979998167, 0.256039249999958, 0.2549249769999733, 0.26770503499983533, 0.2585111280000092, 0.2534973199999513, 0.2574933930000043, 0.2688835730000392, 0.25709193600005165, 0.2582924620000995, 0.2551701239999602, 0.2579445010001109, 0.2537198909999461, 0.258160615999941, 0.257400453000173, 0.2575714600000083, 0.25719076500013216, 0.25602168299997174, 0.2612546560001192, 0.2569049520000135, 0.25658759199995984, 0.25826334699991094, 0.25390822100007426, 0.25522967400002017, 0.25978250799994385, 0.2568119569998544, 0.2612774450001325, 0.2602062579999256, 0.26707800600001974, 0.25654055499990136, 0.25464717400018344, 0.2567932940000901, 0.2616354109998156, 0.25789041799998813, 0.26002433899998323, 0.25881536300016705, 0.25908980100007284, 0.2625190040000689, 0.2601776740000332]
Number of tests: 50
The speed of "import sympy" is: 0.258233 +- 0.003504
eendebakpt@woelmuis:~/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.2627209590000348, 0.2606937199998356, 0.25977801800013367, 0.2668773880000117, 0.25525705599989124, 0.2595361710000361, 0.2584197160001622, 0.25180167500002426, 0.25751827199997024, 0.25490321699999186, 0.2551755930001036, 0.2596296629999415, 0.2635841750000054, 0.2573761200001172, 0.2553685039999891, 0.2559703060001084, 0.25835445799998524, 0.2524185590000343, 0.2570603270000902, 0.25519752699983655, 0.2634718380002141, 0.2549482720000924, 0.25889327100003356, 0.25680885099995976, 0.2593443959999604, 0.2589607429999887, 0.252508078000119, 0.25633841100011523, 0.26284038900007545, 0.25504915600004097, 0.2614197939999485, 0.25328517200000533, 0.25799999299988485, 0.25643479199993635, 0.2616250169999148, 0.2559688650001135, 0.2564684930000567, 0.26460269800008973, 0.2579310290000194, 0.2577397129998644, 0.2574612999999317, 0.2559228080001503, 0.2592914419999488, 0.2607275979999031, 0.2515977670000211, 0.256944552999812, 0.258465249999972, 0.2566703930001495, 0.2568872580000061, 0.2558636070000375, 0.25956973699999253]
Number of tests: 50
The speed of "import sympy" is: 0.257739 +- 0.003201

Other comments

The registration inside matmul and matadd is a bit special. The methods registered there are ambiguous, but this is accepted by passing a special ambiguity_register_error_ignore_dup as the on_ambiguity argument. For this reason we temporary enable the ordering when registering inside these modules.

Release Notes

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<!-- Your title above should be a short description of what
was changed. Do not include the issue number in the title. -->

#### Brief description of what is fixed or changed

During the sympy import several Dispatchers are created and methods are registered. On each registration the full set of methods is ordered, which is costly. 

This PR reduces the sorting by

1) Stopping the sorting before importing subpackages
2) After importing subpackges (which includes the registration of methods to dispatchers), sort all dispatchers

Existing functionality of the `Dispatcher` is used.

The import time is reduced from 269 to 258 ms.

<details>
<summary>Output of bin/test_import</summary>

Master: 
```
eendebakpt@woelmuis:/mnt/data/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.2750202629999876, 0.27497193600015635, 0.2702798200000416, 0.26889375400014615, 0.2687653349998982, 0.2716369390000182, 0.2794790110001486, 0.26655530899984115, 0.2691565279999395, 0.2684384439999121, 0.27359299299996565, 0.26817051800003355, 0.26997147999986737, 0.26848315500001263, 0.2695972559999973, 0.2689053330000206, 0.266499002999808, 0.27336413400007586, 0.2695815440001752, 0.2726911500001279, 0.2669147869999051, 0.26745493099997475, 0.27697483599990846, 0.26868902599994726, 0.2712855639999816, 0.26795200600008684, 0.268541746999972, 0.26775163699994664, 0.27113610199990035, 0.2674677069999234, 0.26795610500016664, 0.26821584900017115, 0.26595949700003985, 0.2713385689999086, 0.2666728149999926, 0.2681342289999975, 0.2687668170001416, 0.2691814659999636, 0.26562634600009005, 0.26544889799993143, 0.2664543210000829, 0.2697006410000995, 0.2687029410001287, 0.26908488100002614, 0.2688644029999523, 0.2712997159999304, 0.27255410699990534, 0.2704326430000492, 0.270944481000015, 0.2717818040000566, 0.2686064910001278]
Number of tests: 50
The speed of "import sympy" is: 0.269579 +- 0.002746
eendebakpt@woelmuis:/mnt/data/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.27012507099993854, 0.2690900089999104, 0.2714201490000505, 0.266401506999955, 0.269292661999998, 0.2699106360000769, 0.2750912349999908, 0.2690173099999811, 0.2662025030001587, 0.27493229600008817, 0.2732122560000789, 0.26986700499992367, 0.2673109410000052, 0.2685452460000306, 0.2705744620000132, 0.27552572399986275, 0.26664292999998906, 0.2703416899998956, 0.27229860700003883, 0.2681302329999653, 0.26659810000001016, 0.2696212359999208, 0.2680760770001598, 0.2669595599998047, 0.2689367319999292, 0.2691656189999776, 0.2738177270000506, 0.270048332999977, 0.27325413200014737, 0.26922133700009, 0.2687775770000371, 0.2711154799999349, 0.2729384140000093, 0.2654830999999831, 0.270534656999871, 0.27074404199993296, 0.2697775599999659, 0.2736168829999315, 0.26733127000011336, 0.2783130220000203, 0.2690745869999773, 0.2671201389998714, 0.26748783299990464, 0.2705764470001668, 0.26714791799986415, 0.26750925000010284, 0.2674039340001855, 0.2647815189998255, 0.2660385930000757, 0.2756670290000329, 0.26935999700003777]
Number of tests: 50
The speed of "import sympy" is: 0.269806 +- 0.002957
```

PR
```
eendebakpt@woelmuis:~/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.25928017599994746, 0.25534695700002885, 0.25327333200016255, 0.2605424830001084, 0.25428364799995506, 0.25779407999993964, 0.25567224200017336, 0.255802980999988, 0.2575704589999077, 0.2656662080000842, 0.25502123999990545, 0.25856492699995215, 0.2610679979998167, 0.256039249999958, 0.2549249769999733, 0.26770503499983533, 0.2585111280000092, 0.2534973199999513, 0.2574933930000043, 0.2688835730000392, 0.25709193600005165, 0.2582924620000995, 0.2551701239999602, 0.2579445010001109, 0.2537198909999461, 0.258160615999941, 0.257400453000173, 0.2575714600000083, 0.25719076500013216, 0.25602168299997174, 0.2612546560001192, 0.2569049520000135, 0.25658759199995984, 0.25826334699991094, 0.25390822100007426, 0.25522967400002017, 0.25978250799994385, 0.2568119569998544, 0.2612774450001325, 0.2602062579999256, 0.26707800600001974, 0.25654055499990136, 0.25464717400018344, 0.2567932940000901, 0.2616354109998156, 0.25789041799998813, 0.26002433899998323, 0.25881536300016705, 0.25908980100007284, 0.2625190040000689, 0.2601776740000332]
Number of tests: 50
The speed of "import sympy" is: 0.258233 +- 0.003504
eendebakpt@woelmuis:~/sympy$ bin/test_import 
Note: the first run (warm up) was not included in the average + std dev
All runs (including warm up):
[0.2627209590000348, 0.2606937199998356, 0.25977801800013367, 0.2668773880000117, 0.25525705599989124, 0.2595361710000361, 0.2584197160001622, 0.25180167500002426, 0.25751827199997024, 0.25490321699999186, 0.2551755930001036, 0.2596296629999415, 0.2635841750000054, 0.2573761200001172, 0.2553685039999891, 0.2559703060001084, 0.25835445799998524, 0.2524185590000343, 0.2570603270000902, 0.25519752699983655, 0.2634718380002141, 0.2549482720000924, 0.25889327100003356, 0.25680885099995976, 0.2593443959999604, 0.2589607429999887, 0.252508078000119, 0.25633841100011523, 0.26284038900007545, 0.25504915600004097, 0.2614197939999485, 0.25328517200000533, 0.25799999299988485, 0.25643479199993635, 0.2616250169999148, 0.2559688650001135, 0.2564684930000567, 0.26460269800008973, 0.2579310290000194, 0.2577397129998644, 0.2574612999999317, 0.2559228080001503, 0.2592914419999488, 0.2607275979999031, 0.2515977670000211, 0.256944552999812, 0.258465249999972, 0.2566703930001495, 0.2568872580000061, 0.2558636070000375, 0.25956973699999253]
Number of tests: 50
The speed of "import sympy" is: 0.257739 +- 0.003201
```
</details>

#### Other comments

The registration inside `matmul` and `matadd` is a bit special. The methods registered there are ambiguous, but this is accepted by passing a special `ambiguity_register_error_ignore_dup` as the `on_ambiguity` argument. 
For this reason we temporary enable the ordering when registering inside these modules.

#### Release Notes

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* solvers
  * Added a new solver for logarithmic equations.

* functions
  * Fixed a bug with log of integers.

or if no release note(s) should be included use:

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[github-actions[bot]]

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)

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

[oscarbenjamin]

Would this work if SymPy was using the PyPI multipledispatch module? I'd really like to get rid of SymPy's multipledispatch module and use that instead.

[eendebakpt]

Would this work if SymPy was using the PyPI multipledispatch module? I'd really like to get rid of SymPy's multipledispatch module and use that instead.

The pypi multipledispatch and sympy multipledispatch are obviously related. But there are differences, compare for example https://github.com/mrocklin/multipledispatch/blob/master/multipledispatch/conflict.py#L9 with https://github.com/sympy/sympy/blob/master/sympy/multipledispatch/conflict.py#L7. Replacing the dispatchers with the pypi dispatcher is not something I will pick up.

[oscarbenjamin]

Replacing the dispatchers with the pypi dispatcher is not something I will pick up.

Fair enough but my point is that I don't want to introduce any changes that would be incompatible with the other multipledispatch module.

[asmeurer]

It doesn't look like this changes anything in multipledispatch, so I don't see that being an issue.

[asmeurer]

Although these are apparently deprecated upstream https://github.com/mrocklin/multipledispatch/blob/master/multipledispatch/dispatcher.py#L30-L47.

[asmeurer]

I guess upstream made the lazy behavior the default. I wonder if it is faster than this.

This change looks good for now, though.

[bjodah]

Personally, I'm no fan of having pairs of calls at module level that shares and mutates global state. I would at least like to see such pairs wrapped in a context manager.

[oscarbenjamin]

I opened #23830 to consider the possibility of using the pypi multipledispatch module.