bpo-41513: Expand comments and add references for a better understanding (GH-22123)
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@ -2419,9 +2419,9 @@ To avoid overflow/underflow and to achieve high accuracy giving results
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that are almost always correctly rounded, four techniques are used:
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* lossless scaling using a power-of-two scaling factor
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* accurate squaring using Veltkamp-Dekker splitting
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* compensated summation using a variant of the Neumaier algorithm
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* differential correction of the square root
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* accurate squaring using Veltkamp-Dekker splitting [1]
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* compensated summation using a variant of the Neumaier algorithm [2]
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* differential correction of the square root [3]
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The usual presentation of the Neumaier summation algorithm has an
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expensive branch depending on which operand has the larger
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@ -2456,7 +2456,11 @@ Given that csum >= 1.0, we have:
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Since lo**2 is less than 1/2 ulp(csum), we have csum+lo*lo == csum.
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To minimize loss of information during the accumulation of fractional
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values, each term has a separate accumulator.
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values, each term has a separate accumulator. This also breaks up
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sequential dependencies in the inner loop so the CPU can maximize
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floating point throughput. [5] On a 2.6 GHz Haswell, adding one
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dimension has an incremental cost of only 5ns -- for example when
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moving from hypot(x,y) to hypot(x,y,z).
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The square root differential correction is needed because a
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correctly rounded square root of a correctly rounded sum of
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@ -2466,7 +2470,7 @@ The differential correction starts with a value *x* that is
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the difference between the square of *h*, the possibly inaccurately
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rounded square root, and the accurately computed sum of squares.
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The correction is the first order term of the Maclaurin series
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expansion of sqrt(h**2 + x) == h + x/(2*h) + O(x**2).
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expansion of sqrt(h**2 + x) == h + x/(2*h) + O(x**2). [4]
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Essentially, this differential correction is equivalent to one
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refinement step in Newton's divide-and-average square root
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@ -2474,12 +2478,24 @@ algorithm, effectively doubling the number of accurate bits.
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This technique is used in Dekker's SQRT2 algorithm and again in
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Borges' ALGORITHM 4 and 5.
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Without proof for all cases, hypot() cannot claim to be always
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correctly rounded. However for n <= 1000, prior to the final addition
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that rounds the overall result, the internal accuracy of "h" together
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with its correction of "x / (2.0 * h)" is at least 100 bits. [6]
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Also, hypot() was tested against a Decimal implementation with
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prec=300. After 100 million trials, no incorrectly rounded examples
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were found. In addition, perfect commutativity (all permutations are
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exactly equal) was verified for 1 billion random inputs with n=5. [7]
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References:
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1. Veltkamp-Dekker splitting: http://csclub.uwaterloo.ca/~pbarfuss/dekker1971.pdf
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2. Compensated summation: http://www.ti3.tu-harburg.de/paper/rump/Ru08b.pdf
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3. Square root differential correction: https://arxiv.org/pdf/1904.09481.pdf
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4. https://www.wolframalpha.com/input/?i=Maclaurin+series+sqrt%28h**2+%2B+x%29+at+x%3D0
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5. https://bugs.python.org/file49439/hypot.png
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6. https://bugs.python.org/file49435/best_frac.py
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7. https://bugs.python.org/file49448/test_hypot_commutativity.py
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*/
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