gh-117999: Fix small integer powers of complex numbers by serhiy-storchaka · Pull Request #124243 · python/cpython · GitHub

Fix the sign of zero components in the result. E.g. complex(1,-0.0)**2 now evaluates to complex(1,-0.0) instead of complex(1,0.0).
Fix negative small integer powers of infinite complex numbers. E.g. complex(inf)**-1 now evaluates to complex(0,-0.0) instead of complex(nan,nan).
Powers of infinite numbers no longer raise OverflowError. E.g. complex(inf)**1 now evaluates to complex(inf) and complex(inf)**0.5 now evaluates to complex(inf,nan).

Issue: Small integer powers of real±0j are invalid #117999

* Fix the sign of zero components in the result. E.g. complex(1,-0.0)**2 now evaluates to complex(1,-0.0) instead of complex(1,-0.0). * Fix negative small integer powers of infinite complex numbers. E.g. complex(inf)**-1 now evaluates to complex(0,-0.0) instead of complex(nan,nan). * Powers of infinite numbers no longer raise OverflowError. E.g. complex(inf)**1 now evaluates to complex(inf) and complex(inf)**0.5 now evaluates to complex(inf,nan).

I'll take look, but probably already tomorrow :(

Meanwhile, I'm not sure if fix for division should be included here. That does make sense for a separate pr, that will provide mixed-mode arithmetic for complex.

I was thinking (In fact, I did, patch just lacks more tests: skirpichev#4) on splitting out from skirpichev#1 just cases for real op complex, as specified by C99+ (note, that there is no special case for real/complex per standard). That's something unfortunate, but imaginary numbers gone from latest drafts of upcoming C standard.

That looks close to #118000. I doubt there is an easy way to fix powi() algorithm to match generic one (same signs, etc).

In mathematics, the non-integer power of a complex number is a multi-value function. We can choose the main value, but this is arbitrary. The integer power of a complex number is always defined unambiguously. It is also more useful. So it is expected that the integer power can be calculated with higher precision and produce less exceptions or NaNs that the general real or complex power.

We can choose the main value, but this is arbitrary.

No, this will introduce a huge incompatibility, not just with the C standard. z**a (when a single-valued expected) is usually defined as Exp[a Log[z]], where the principal value Log[z] is taken, e.g. as in Mathematica.

So it is expected that the integer power can be calculated with higher precision and produce less exceptions or NaNs

I doubt you can avoid nan component for complex('inf')**2 with the current algorithm.

It turns to be same as #118000, modulo changes in _Py_ADJUST_ERANGE2. I stuck with examples like z=1-1j, but probably we can't do anything in such cases.

Now it looks good for me.

Few remarks:

See suggestion.
_Py_ADJUST_ERANGE2 change looks incomplete for me. I think proper fix should include special case for real/integer exponents, just like we did for arithmetic (I'll open an issue.) But if you wish to keep this, lets add tests.

For me, your tests works fine with my patch from #118000. Here it is, with slightly adjusted formatting (and using new _Py_rc_quot):

diff --git a/Objects/complexobject.c b/Objects/complexobject.c
index b66ebe131a..9dc5d22b37 100644
--- a/Objects/complexobject.c
+++ b/Objects/complexobject.c
@@ -333,19 +333,26 @@ _Py_c_pow(Py_complex a, Py_complex b)
         r.real = len*cos(phase);
         r.imag = len*sin(phase);
 
-        _Py_ADJUST_ERANGE2(r.real, r.imag);
+        if (isfinite(a.real) && isfinite(a.imag)
+            && isfinite(b.real) && isfinite(b.imag))
+        {
+            _Py_ADJUST_ERANGE2(r.real, r.imag);
+        }
     }
     return r;
 }
 
+#define INT_EXP_CUTOFF 100
+
 static Py_complex
 c_powu(Py_complex x, long n)
 {
-    Py_complex r, p;
+    Py_complex p = x, r = n-- ? p : c_1;
     long mask = 1;
-    r = c_1;
-    p = x;
-    while (mask > 0 && n >= mask) {
+    assert(-1 <= n);
+    assert(n < INT_EXP_CUTOFF);
+    while (n >= mask) {
+        assert(mask > 0);
         if (n & mask)
             r = _Py_c_prod(r,p);
         mask <<= 1;
@@ -357,11 +364,10 @@ c_powu(Py_complex x, long n)
 static Py_complex
 c_powi(Py_complex x, long n)
 {
-    if (n > 0)
-        return c_powu(x,n);
+    if (n >= 0)
+        return c_powu(x, n);
     else
-        return _Py_c_quot(c_1, c_powu(x,-n));
-
+        return _Py_rc_quot(1.0, c_powu(x, -n));
 }
 
 double
@@ -751,9 +757,13 @@ complex_pow(PyObject *v, PyObject *w, PyObject *z)
     errno = 0;
     // Check whether the exponent has a small integer value, and if so use
     // a faster and more accurate algorithm.
-    if (b.imag == 0.0 && b.real == floor(b.real) && fabs(b.real) <= 100.0) {
+    if (b.imag == 0.0 && b.real == floor(b.real)
+        && fabs(b.real) <= INT_EXP_CUTOFF)
+    {
         p = c_powi(a, (long)b.real);
-        _Py_ADJUST_ERANGE2(p.real, p.imag);
+        if (isfinite(a.real) && isfinite(a.imag)) {
+            _Py_ADJUST_ERANGE2(p.real, p.imag);
+        }
     }
     else {
         p = _Py_c_pow(a, b);

(asserts are outcome of review, I'm not sure they are useful here.)

What do you think?

CC @mdickinson
CC @picnixz

FYI, issue for pow(): #137746

This looks good to me though I have a small suggestion.

Took me a while to remember that this was 1 + 0j (maybe rename that variable in a separate issue, e.g., ONE_AS_COMPLEX).

This now used just in one place. If we need this branch - it's better to inline struct value here.

If this is the case, then yes, let's inline it.

If you want to make this function a bit more natural, I would suggest having if (n == 0) { return c_1; } and remove the assertion n > 0 and just assume n is an unsigned long (or change the assertion to n >= 0). I don't think it's necessary to handle the case n == 0 in c_powi().

See my patch above: #124243 (comment)

Oh I remember this one. I think we can just keep special-casing n == 0 without actually wondering about the cutoff limit. A generic square-and-multiply algorithm is fine IMO.

Well, it's possible to find base and exponent, when binary exponentiation is worse.

E.g. on this branch, if we omit cutoff, on my system I have:

$ ./python -m timeit -u nsec -s 'z=1e-10+1j;e=1e18+0j' 'z**e'
200000 loops, best of 5: 1.39e+03 nsec per loop
$ ./python -m timeit -u nsec -s 'z=1e-10+1j;e=1e18+0j;from _testcapi import _py_c_pow as p' 'p(z, e)'
200000 loops, best of 5: 1.12e+03 nsec per loop

(Though, both results looks to be a garbage.)

Co-authored-by: Serhiy Storchaka <storchaka@gmail.com>

This PR is stale because it has been open for 30 days with no activity.

Co-authored-by: Sergey B Kirpichev <skirpichev@gmail.com>

Added cosmetic changes form #118000 (inlining c_1 and INT_EXP_CUTOFF).

…H-124243) * Fix the sign of zero components in the result. E.g. complex(1,-0.0)**2 now evaluates to complex(1,-0.0) instead of complex(1,-0.0). * Fix negative small integer powers of infinite complex numbers. E.g. complex(inf)**-1 now evaluates to complex(0,-0.0) instead of complex(nan,nan). * Powers of infinite numbers no longer raise OverflowError. E.g. complex(inf)**1 now evaluates to complex(inf) and complex(inf)**0.5 now evaluates to complex(inf,nan).

serhiy-storchaka requested a review from skirpichev September 19, 2024 13:31

bedevere-app Bot added the awaiting core review label Sep 19, 2024

bedevere-app Bot mentioned this pull request Sep 19, 2024

Small integer powers of real±0j are invalid #117999

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serhiy-storchaka mentioned this pull request Sep 19, 2024

gh-117999: use generic algorithm in complex_pow() if base has special components #123283

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skirpichev reviewed Sep 20, 2024

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Comment thread Lib/test/test_complex.py Outdated

Comment thread Objects/complexobject.c Outdated

Comment thread Objects/complexobject.c

Comment thread Objects/complexobject.c

skirpichev mentioned this pull request Oct 2, 2024

gh-69639: add mixed-mode rules for complex arithmetic (C-like) #124829

Merged

serhiy-storchaka added 5 commits November 26, 2024 20:26

Merge branch 'main' into complex_powi

5b5320a

Polish tests.

1304423

Remove unrelated change.

bc4adfd

Merge branch 'main' into complex_powi

4b8535d

Add more tests.

e876273

skirpichev self-requested a review February 4, 2025 01:34

Merge branch 'main' into complex_powi

bb748c7

skirpichev approved these changes Aug 13, 2025

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skirpichev requested a review from picnixz August 13, 2025 10:23

picnixz approved these changes Aug 18, 2025

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bedevere-app Bot added awaiting merge and removed awaiting core review labels Aug 18, 2025

skirpichev added a commit to skirpichev/cpython that referenced this pull request Sep 17, 2025

take tests from python#124243

9e9b454

Co-authored-by: Serhiy Storchaka <storchaka@gmail.com>

github-actions Bot added the stale Stale PR or inactive for long period of time. label Apr 18, 2026

skirpichev added 2 commits August 18, 2026 10:33

Merge branch 'main' into complex_powi

3090707

fix test (smaller eps value to pass assertClose tests)

c4cd2e7

skirpichev mentioned this pull request Aug 19, 2026

gh-117999: fixed small integer powers of complex numbers #118000

Closed

github-actions Bot removed the stale Stale PR or inactive for long period of time. label Aug 22, 2026

skirpichev mentioned this pull request Aug 31, 2026

gh-156693: correct complex powers with infinite bases #156694

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serhiy-storchaka and others added 2 commits September 3, 2026 10:42

Merge branch 'main' into complex_powi

6b3e781

Remove the c_1 constant and add the INT_EXP_CUTOFF macro.

fcfa3f6

Co-authored-by: Sergey B Kirpichev <skirpichev@gmail.com>

serhiy-storchaka merged commit d56f7e2 into python:main Sep 3, 2026
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serhiy-storchaka deleted the complex_powi branch September 3, 2026 08:43

serhiy-storchaka mentioned this pull request Sep 3, 2026

Incorrect overflows for complex powers when base is infinite #156693

Closed

+                  assert(n > 0);
+                  while ((n & 1) == 0) {
+                      x = _Py_c_prod(x, x);
+                      n >>= 1;
+                  }
+                  Py_complex r = x;
+                  n >>= 1;
+                  while (n) {
+                      x = _Py_c_prod(x, x);
+                      if (n & 1) {
+                          r = _Py_c_prod(r, x);
+                      }
+                      n >>= 1;

gh-117999: Fix small integer powers of complex numbers by serhiy-storchaka · Pull Request #124243 · python/cpython · GitHub

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gh-117999: Fix small integer powers of complex numbers - #124243

gh-117999: Fix small integer powers of complex numbers#124243serhiy-storchaka merged 11 commits intopython:mainpython/cpython:mainfrom serhiy-storchaka:complex_powiserhiy-storchaka/cpython:complex_powiCopy head branch name to clipboard

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gh-117999: Fix small integer powers of complex numbers#124243
serhiy-storchaka merged 11 commits into
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