gh-140443: Use fma in loghelper to improve accuracy of log for very large integers (#140469)

* gh-140443:use fma in loghelper to improve accuracy of log for very large integers

Use fused multiply-add in log_helper() for huge ints.

Saving a rounding here is remarkably effective. Across some millions
of randomized test cases with ints up to a billion bits, on Windows
and using log10, the ULP error distribution was dramatically
flattened, and its range was nearly cut in half. In fact, the largest
error Tim saw was under 0.6 ULP.

---------

Co-authored-by: abhi210 <[email protected]>
Co-authored-by: Stan Ulbrych <[email protected]>

Author: Abhi210

Misc/ACKS

@@ -1921,6 +1921,7 @@ Tim Tisdall
 Jason Tishler
 Christian Tismer
 Jim Tittsler
+Abhishek Tiwari
 Frank J. Tobin
 James Tocknell
 Bennett Todd

Misc/NEWS.d/next/Core_and_Builtins/2025-10-22-23-26-37.gh-issue-140443.wT5i1A.rst

@@ -0,0 +1,5 @@
+The logarithm functions (such as :func:`math.log10` and :func:`math.log`) may now produce
+slightly different results for extremely large integers that cannot be
+converted to floats without overflow. These results are generally more
+accurate, with reduced worst-case error and a tighter overall error
+distribution.

Modules/mathmodule.c

@@ -2309,7 +2309,7 @@ loghelper(PyObject* arg, double (*func)(double))
             assert(e >= 0);
             assert(!PyErr_Occurred());
             /* Value is ~= x * 2**e, so the log ~= log(x) + log(2) * e. */
-            result = func(x) + func(2.0) * e;
+            result = fma(func(2.0), (double)e, func(x));
         }
         else
             /* Successfully converted x to a double. */