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bpo-41513: Expand comments #22123

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Sep 6, 2020
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bpo-41513: Expand comments #22123

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0f7a2ff
Note performance aspects of using multiple accumulators
rhettinger Sep 6, 2020
7a29756
Add accuracy notes
rhettinger Sep 6, 2020
971ae45
Show where each footnote applies
rhettinger Sep 6, 2020
080c68a
Better wording for the incremental loop costs
rhettinger Sep 6, 2020
b6ecce6
Improve wording for the accuracy paragraph
rhettinger Sep 6, 2020
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Add accuracy notes
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@rhettinger
rhettinger committed Sep 6, 2020
commit 7a29756a9d916ee54dc741ff23e00a8d95205c74
10 changes: 10 additions & 0 deletions Modules/mathmodule.c
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Expand Up @@ -2477,12 +2477,22 @@ algorithm, effectively doubling the number of accurate bits.
This technique is used in Dekker's SQRT2 algorithm and again in
Borges' ALGORITHM 4 and 5.

Without proof for all cases, hypot() cannot claim to be always
correctly rounded. However, the accuracy of "h + x / (2.0 * h)" for
n <= 1000 is at least 100 bits prior to the final rounding. Also,
hypot() was tested against a Decimal implementation with prec=300.
After 100 million trials no incorrectly rounded examples were found.
In addition, perfect commutativity (all permutations are equal) was
verified for 1 billion random inputs with n=5.

References:

1. Veltkamp-Dekker splitting: http://csclub.uwaterloo.ca/~pbarfuss/dekker1971.pdf
2. Compensated summation: http://www.ti3.tu-harburg.de/paper/rump/Ru08b.pdf
3. Square root differential correction: https://arxiv.org/pdf/1904.09481.pdf
4. https://www.wolframalpha.com/input/?i=Maclaurin+series+sqrt%28h**2+%2B+x%29+at+x%3D0
5. https://bugs.python.org/file49435/best_frac.py
6. https://bugs.python.org/file49448/test_hypot_commutativity.py

*/

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