[tombouctou]

The rest of #2047 step 6, for complex numbers: complex arithmetic, and functions with a complex value at a real argument, now get their precision the way WMA does.

In WMA each part of a Complex keeps its own error. Accuracy combines the two errors in quadrature, and Precision is that accuracy plus Log10 of the modulus; an exact part also counts toward the modulus. Before this PR, Mathics3 gave a complex result the precision of its least precise input, or machine precision, which is why 1.5`20.3 + 2.5`20.3 I had precision 15.

• Plus and Times on complex numbers with exact or PrecisionReal parts work out each part with the real rules. Times multiplies as a balanced tree and squares a pair of equal factors. z^n squares starting from the top bit of n, and 1/z is (a - b I)/(a^2 + b^2). This is the order WMA uses, and it changes the last digits of the precision. 2.`20 I keeps an exact zero real part.
• A function with a complex value at a real argument (ArcSin[2.`20], Log[-1.`20], ArcCosh[020], ArcSec[0.520]``, …) gives each part the error from that part of x f'[x], as #2052–#2056 do for real values. A part that does not change gets 2^-54of the error, so thePi part of ``Log[-1.20]`` is 16 digits more precise than the real part, as in WMA. At a zero or a branch point, the change of f within the error goes to the nonzero parts.
• A negative PrecisionReal to a Rational power: x^(1/q) shares the error of the whole value between its parts, and x^(p/q) is x^n (x^(1/q))^r. Checked against WMA for p/q from −5/2 to 7/3.
• Log[x] is no longer rewritten as Log[E, x]: the numerified E base added an error of its own.
• An inexact zero real part is now shown, as in WMA (019.4 + 2.19.7 I``). N[I, 30] is ``1.30 I``.

Input	master	this PR	WMA 15
Precision[ArcSec[0.5`20]]	15.1196	20.0571	20.0571
Precision[1.5`20.3 + 2.5`20.3 I]	15.	20.3	20.3
ArcCosh[020]``	1.5707963…`20.*I	1.5707963…`20.1961*I	1.5707963…`20.1961*I
Sqrt[-2.`20]	1.4142135…`20.*I	1.4142135…`20.301*I	1.4142135…`20.301*I
Precision[Log[-1.`20]]	15.4971	20.4971	20.4971
Accuracy[Im[Log[-1.`20]]] - Accuracy[Re[Log[-1.`20]]]	4.50285	16.2556	16.2556
Precision[(1.`20 + 2.`25 I) (3.`22 - 1.`18 I)]	15.	18.4939	18.4939
Precision[(1.`20 + 2.`25 I)^5]	15.	19.3049	19.3049
Precision[(-8.`20)^(1/3)]	20.	20.4771	20.4771

Not in this PR: functions of a complex argument (Sqrt[1.`20 + I], Exp[1.`20 I]), a negative base with an inexact exponent, and N of an exact complex number with two nonzero parts, where WMA splits the precision between the parts.

Tests

test_complex_precision and test_complex_valued_precision in test/builtin/atomic/test_numbers.py, next to test_real_valued_precision, with WMA 15 values. Full run on kimmy: all doctests pass. pytest under xdist (-n 4) fails a different handful of tests on each run, on master as well, because those tests depend on symbols that earlier tests in the same worker define or clear (f, F, x). With --dist loadfile, master fails 15 (test_forms.py, test_options.py, test_testing_expressions.py, test_latex.py) and this branch fails 7 (test_list.py, test_diffeqns.py, test_constructing.py). None of them is about numbers, and each of these files passes when run on its own.

Benchmark from #2050 on kimmy, best of 5, ms:

Case	master	this PR
Total[Table[N[1/k, 50], {k, 2000}]]	2645	2738
Times @@ Table[N[1 + 1/k, 50], {k, 2000}]	3255	3324
Do[x = N[k, 50] + Pi, {k, 2000}]	5388	5238
Total[Table[1./k, {k, 2000}]]	1988	2056
Total[Table[N[1/k + I/(k+1), 50], {k, 2000}]]	4884	5127
Times @@ Table[N[1 + I/k, 50], {k, 500}]	883	1070
Table[N[1 + I/k, 30]^5, {k, 500}]	1435	1820
Table[1 / N[1 + I/k, 30], {k, 500}]	1592	1670
Table[ArcSin[N[1 + k, 30]], {k, 500}]	1112	1504
Table[Log[N[-k, 30]], {k, 500}]	1421	1322

The real cases are unchanged within noise. Complex Times, z^5 and ArcSin outside its domain are 20–35% slower: their parts are now computed from real operations, and ArcSin takes more derivatives.

🤖 Generated with Claude Code

[rocky]

LGTM