[tombouctou]

Part of step 6 of the outline in #2047, after #2050: N[expr, p] gives precision p, as WMA does.

Input	master	this PR	WMA 15
Precision[N[Pi, 20.3]]	20.	20.3	20.3
Precision[N[Pi, 41/2]]	20.	20.5	20.5
Precision[N[Pi E, 20]]	19.699	20.	20.
Precision[N[Pi - 314159/100000, 20]]	13.6257	20.	20.
Precision[N[1.5`10, 20.3]]	10.	10.	10.
Precision /@ N[{Pi, 1.5`10, 2.5`30.2}, 20.3]	{20., 10., 20.}	{20.3, 10., 20.3}	{20.3, 10., 20.3}
$MaxExtraPrecision	$MaxExtraPrecision	50.	50.

N[Pi E, 20] giving 19.699 came in with #2050: N makes Pi and E 20-digit numbers, and Times now adds their errors.

Changes

• N[expr, p] (eval_NValues_to_precision in mathics/eval/nevaluator.py) first evaluates at p, as before. If a nonzero arbitrary-precision number in the result has fewer than p digits, it evaluates again with the digits lost added, up to $MaxExtraPrecision more than p. It goes on while some number short of p digits gains digits; one that gains nothing has an input with fewer than p digits, and keeps them. Then every number above p is set to precision p.
• The N rules see the working precision. So a retry is used only if it has the same form as the first result, up to the values of numbers: with N[a, 20] = 11, N[a Pi, 20] must not turn back into a 3.14… at 21 digits. The first pass is still at exactly p, so a rule given for p applies as before.
• $MaxExtraPrecision, new, 50. by default, no attributes. Assigning it anything other than a non-negative number or Infinity gives $MaxExtraPrecision::precset, as in WMA.
• N::meprec when the extra digits are not enough. As in WMA, there is no message when $MaxExtraPrecision is 0.

N[expr, {p, a}] and the internal eval_N are unchanged.

Tests

• test_n_real_valued_precision and test_max_extra_precision in test/builtin/atomic/test_numbers.py, with WMA 15 values.
• Doctests for $MaxExtraPrecision.
• The RootSum doctest N[Integrate[1/(x^5 + 11 x + 1), {x, 1, 3}], 50] expected …857550894. On master that number has precision 49.66, so its last digits were off. Now it is …857553407, as in WMA (…8575534094…; NIntegrate at 70 digits agrees).

Full run on kimmy (Python 3, pytest -n 4 and docpipeline.py), next to master 9f166b1:

• Doctests: 4611 passed, 0 failed (master: 4604; the 7 new ones are for $MaxExtraPrecision).
• pytest: 34 failed on this branch, 58 on master. As noted in PrecisionReal: keep a real-valued precision, as WMA does (#2047) #2050, these depend on the order the tests run in. The 12 that fail only on this branch (test_tensor.py, test_testing_expressions.py) pass when their files run on their own.

Benchmark

kimmy, best of 7, both runs at the same time:

	master	this PR
Total[Table[N[1/k, 50], {k, 2000}]]	2636 ms	2586 ms
Times @@ Table[N[1 + 1/k, 50], {k, 2000}]	3113 ms	3104 ms
Do[x = N[k, 50] + Pi, {k, 2000}]	5512 ms	5277 ms
Total[Table[1./k, {k, 2000}]] (machine reals, control)	1921 ms	1850 ms
Do[N[Pi E + Sqrt[2], 30], {k, 300}]	1118 ms	1643 ms
Do[N[Sin[k] + Cos[k], 30], {k, 300}]	787 ms	935 ms

The benchmark from #2050 doesn't change. An N[expr, p] whose first pass loses digits now costs about one more evaluation (the last two rows): that is the second pass.

Still to do from step 6

• Functions other than Plus and Times (Sqrt, Power, Exp, …): precision from the condition number.
• SetPrecision, SetAccuracy.
• Low-precision inputs stored with few bits (Precision[1.5`20 - 1.4`2]).
• Precision of a list with a machine number.

🤖 Generated with Claude Code

[rocky]

When a new built-in is added, SYMBOLS_MANIFEST.txt should be updated.

You can run admin-tools/build_and_check_manifest.py to automatically update this. (When I just did, I also noticed that XMLXMLObjectImport` needs to be added.)

[tombouctou]

Done in e16f1b4: I reran admin-tools/build_and_check_manifest.py, which adds $MaxExtraPrecision. It also adds PolynomialMod, which was missing on master, and puts TagUnset in order.

About XMLXMLObjectImport: it is already in the manifest, as its last line. The script printed the wrong name. In the loop over new symbols it printed name, a variable left over from the loop before, which ends on the last manifest entry. So each new symbol (here $MaxExtraPrecisionandPolynomialMod) was reported as XMLXMLObjectImport. The same commit makes it print the missing symbol.

[rocky]

Thanks.

[rocky]

As before, I think this is awesome!

[rocky]

LGTM