lazy evaluation of rational powers

[mxhbl]

Rational powers are currently eagerly evaluated, so that, for example,

using SymbolicUtils
@syms x
(2x) ^ (1//2) == 1.414… * x^(1//2)
simplify(term(^, 2, 1//2)) == 1.414…

and so on. While this might be intentional, it often causes precision loss and makes many analytical expressions rather ugly.

This PR adds branches to basicsymbolic and ^ to prevent evaluation of powers if the base is a literal number and the exponent is a rational. Integer or float exponents still get evaluated. With this PR, we therefore have behavior like

(2x) ^ (1//2) ==  2^(1//2) * x^(1//2)
simplify(term(^, 2, 1//2)) == 2^(1//2)
simplify(term(^, 1//3, 2)) == 1//9
simplify(term(^, 2, 0.5)) == 1.414…

However, there is a small quirk. Currently (2x) ^ 2 results in (4//1) * x^2, i.e. the integer prefactor is converted to a rational, due to the behavior of the unstable_pow function. I am not sure if this is intentional and I did not follow this conversion in the code in this PR. This means with this PR and a rational exponent, we have (2x) ^ (2//1) == 4 * x^(2//1). I guess if this PR gets merged, this behavior should be made consistent, one way or another.

I am curious if this all sounds like a reasonable change, or if there other reasons why immediate evaluation of all powers is preferable — please let me know what you think!

[github-actions[bot]]

Benchmark Results

	master	e06f2c78ec761e...	master/e06f2c78ec761e...
overhead/acrule/a+2	0.728 ± 0.016 μs	0.757 ± 0.019 μs	0.962
overhead/acrule/a+2+b	0.698 ± 0.013 μs	0.725 ± 0.017 μs	0.962
overhead/acrule/a+b	0.251 ± 0.0089 μs	0.258 ± 0.0097 μs	0.975
overhead/acrule/noop:Int	25.1 ± 0.92 ns	26.2 ± 0.91 ns	0.955
overhead/acrule/noop:Sym	0.0365 ± 0.005 μs	0.0368 ± 0.0054 μs	0.992
overhead/rule/noop:Int	0.0441 ± 0.0012 μs	0.0443 ± 0.00084 μs	0.997
overhead/rule/noop:Sym	0.0549 ± 0.0023 μs	0.0548 ± 0.003 μs	1
overhead/rule/noop:Term	0.0547 ± 0.0022 μs	0.055 ± 0.003 μs	0.994
overhead/ruleset/noop:Int	0.128 ± 0.0042 μs	0.129 ± 0.0032 μs	0.99
overhead/ruleset/noop:Sym	0.148 ± 0.0041 μs	0.16 ± 0.0066 μs	0.927
overhead/ruleset/noop:Term	3.12 ± 0.12 μs	3.09 ± 0.1 μs	1.01
overhead/simplify/noop:Int	0.144 ± 0.004 μs	0.14 ± 0.0032 μs	1.03
overhead/simplify/noop:Sym	0.158 ± 0.0055 μs	0.159 ± 0.0065 μs	0.995
overhead/simplify/noop:Term	0.0354 ± 0.0024 ms	0.0378 ± 0.0022 ms	0.936
overhead/simplify/randterm (+, *):serial	0.0838 ± 0.0012 s	0.0897 ± 0.00082 s	0.935
overhead/simplify/randterm (+, *):thread	0.0503 ± 0.03 s	0.0518 ± 0.029 s	0.97
overhead/simplify/randterm (/, *):serial	0.209 ± 0.0067 ms	0.223 ± 0.0078 ms	0.939
overhead/simplify/randterm (/, *):thread	0.236 ± 0.0075 ms	0.254 ± 0.0088 ms	0.929
overhead/substitute/a	0.0582 ± 0.0014 ms	0.0576 ± 0.0014 ms	1.01
overhead/substitute/a,b	0.0511 ± 0.0015 ms	0.0515 ± 0.0014 ms	0.993
overhead/substitute/a,b,c	16.9 ± 0.61 μs	17.6 ± 0.73 μs	0.963
polyform/easy_iszero	28.7 ± 1.7 μs	29.5 ± 1.7 μs	0.972
polyform/isone	2.79 ± 0.01 ns	2.79 ± 0.01 ns	0.997
polyform/iszero	1.11 ± 0.03 ms	1.12 ± 0.032 ms	0.991
polyform/simplify_fractions	1.61 ± 0.043 ms	1.6 ± 0.044 ms	1.01
time_to_load	2.1 ± 0.01 s	2.13 ± 0.023 s	0.988

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR. Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[mxhbl]

Done! Let me know if there is anything else.

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Codecov Report

All modified and coverable lines are covered by tests :white_check_mark:

Project coverage is 82.07%. Comparing base (0288135) to head (2daeb7e). Report is 3 commits behind head on master.

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Additional details and impacted files

@@            Coverage Diff             @@
##           master     #641      +/-   ##
==========================================
+ Coverage   81.86%   82.07%   +0.20%     
==========================================
  Files          16       16              
  Lines        1908     1919      +11     
==========================================
+ Hits         1562     1575      +13     
+ Misses        346      344       -2     

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