[tombouctou]

Part of step 6 of the outline in #2047, after #2051 (merged). The PRs for Log (#2053), the trigonometric and hyperbolic functions (#2054–#2056), SetPrecision (#2057) and low-precision values (#2060) are stacked on this one.

Power and Sqrt get their precision from the condition number, as WMA does. The relative error of b^e is Abs[e] times that of b, plus Abs[e Log[b]] times that of e.

Input	master / #2051	this PR	WMA 15
Precision[Sqrt[1.5`20]]	20.	20.301	20.301
Precision[1.5`20^3]	20.	19.5229	19.5229
Precision[1.5`20^1000]	20.	17.	17.
Precision[1.5`20^(10^-5)]	20.	25.	25.
Precision[2^1.5`20]	20.	19.9831	19.9831
Precision[10^100.`20]	20.	17.6378	17.6378
Precision[1.5`20^2.5`20]	20.	19.4542	19.4542
Precision[Exp[1.5`20]]	20.	19.8239	19.8239

Changes

• MPMathFunction.condition_numbers (mathics/core/builtin.py), new and optional. Given the arguments and the value as mpmath numbers, it returns the factor Abs[x D[f, x] / f] for each argument. The follow-up PRs give it to Log, the trigonometric and the hyperbolic functions; each of those PRs only adds the formulas.
• eval_mpmath_function_by_condition (mathics/eval/numeric.py) uses it when the arguments are exact reals or nonzero PrecisionReals and the value is a nonzero real. The relative errors add up, as in Times from PrecisionReal: keep a real-valued precision, as WMA does (#2047) #2050. When the result has more digits than the arguments (Sqrt, x^(10^-5)), the value is computed again with enough bits. Everything else (complex results, machine numbers, zeros) takes the path it took before.
• Power gives (Abs[e], Abs[e Log[b]]). The factors are typed as mpmath.mpf, not floats: a tiny one would underflow to 0.. Sqrt[x] is x^(1/2), and Exp[x] is E^x.
• eval_N (the internal N) now evaluates as the N builtin does since N[expr, p]: precision exactly p, with extra digits as needed, as in WMA #2051, without the message. Inside N, the 1/2 of Sqrt[2] becomes 0.5 with the requested digits. With this PR its error counts, so eval_N[Sqrt[2], $MachinePrecision] came out with precision 15.07. N gives 15.9546, as WMA does. Two cases in test_evaluators.py now expect the WMA value of N[Sqrt[2], $MachinePrecision] and N[Pi, $MachinePrecision].

Not changed: WMA counts an error for Pi in Pi^1.5`20 (19.4925), but not for E in E^1.5`20. Since #2050, exact constants add no error, so this PR gives 19.7652.

Tests

• test_power_real_valued_precision in test/builtin/atomic/test_numbers.py, with WMA 15 values, and the digits of two results.

Full run on kimmy (pytest -n 4 and docpipeline.py):

• Doctests: all pass.
• pytest: the failures are the ones that depend on the order the tests run in (contexts, packages, patterns, rules, …). Their number changes from run to run. The failing files that touch numbers (test_gamma.py) pass on their own.

Benchmark

kimmy, best of 7, this PR next to #2051, both runs at the same time:

	#2051	this PR
Total[Table[N[1/k, 50], {k, 2000}]]	2687 ms	2564 ms
Times @@ Table[N[1 + 1/k, 50], {k, 2000}]	3187 ms	3049 ms
Do[x = N[k, 50] + Pi, {k, 2000}]	5492 ms	5201 ms
Total[Table[1./k, {k, 2000}]] (machine reals)	1897 ms	1820 ms
Do[Sqrt[N[k, 50]], {k, 2000}]	5412 ms	5188 ms
Do[N[k, 50]^3, {k, 2000}]	4229 ms	4038 ms
Do[2^N[k, 50], {k, 2000}]	4263 ms	4034 ms

No slowdown.

🤖 Generated with Claude Code