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bpo-41513: Add docs and tests for hypot() (GH-22238)

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Raymond Hettinger 2020-09-13 23:33:41 -07:00 • committed by GitHub
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Modules/mathmodule.c
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@ -2429,7 +2429,7 @@ magnitude. We avoid this cost by arranging the calculation so that
fabs(csum) is always as large as fabs(x).
To establish the invariant, *csum* is initialized to 1.0 which is
always larger than x**2 after scaling or division by *max*.
always larger than x**2 after scaling or after division by *max*.
After the loop is finished, the initial 1.0 is subtracted out for a
net zero effect on the final sum. Since *csum* will be greater than
1.0, the subtraction of 1.0 will not cause fractional digits to be
@ -2458,7 +2458,7 @@ Since lo**2 is less than 1/2 ulp(csum), we have csum+lo*lo == csum.
To minimize loss of information during the accumulation of fractional
values, each term has a separate accumulator. This also breaks up
sequential dependencies in the inner loop so the CPU can maximize
floating point throughput. [5] On a 2.6 GHz Haswell, adding one
floating point throughput. [4] On a 2.6 GHz Haswell, adding one
dimension has an incremental cost of only 5ns -- for example when
moving from hypot(x,y) to hypot(x,y,z).
@ -2470,7 +2470,7 @@ The differential correction starts with a value *x* that is
the difference between the square of *h*, the possibly inaccurately
rounded square root, and the accurately computed sum of squares.
The correction is the first order term of the Maclaurin series
expansion of sqrt(h**2 + x) == h + x/(2*h) + O(x**2). [4]
expansion of sqrt(h**2 + x) == h + x/(2*h) + O(x**2). [5]
Essentially, this differential correction is equivalent to one
refinement step in Newton's divide-and-average square root
@ -2492,10 +2492,10 @@ 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/file49439/hypot.png
6. https://bugs.python.org/file49435/best_frac.py
7. https://bugs.python.org/file49448/test_hypot_commutativity.py
4. Data dependency graph: https://bugs.python.org/file49439/hypot.png
5. https://www.wolframalpha.com/input/?i=Maclaurin+series+sqrt%28h**2+%2B+x%29+at+x%3D0
6. Analysis of internal accuracy: https://bugs.python.org/file49435/best_frac.py
7. Commutativity test: https://bugs.python.org/file49448/test_hypot_commutativity.py
*/