bpo-41513: Improve order of adding fractional values. Improve variable names.

[rhettinger]

Improve the variable names and optimize the order that the fractional components are added together. Gives a slight improvement in speed and a minor improvement in accuracy.

Baseline timing:

$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e)'
5000000 loops, best of 11: 83.6 nsec per loop
$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e, tau)'
5000000 loops, best of 11: 89.1 nsec per loop
$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e, tau, e)'
5000000 loops, best of 11: 98.4 nsec per loop

Timing with PR:

$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e)'
5000000 loops, best of 11: 83.1 nsec per loop
$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e, tau)'
5000000 loops, best of 11: 89.1 nsec per loop
$ pytime -s 'from math import pi, hypot, e, tau' 'hypot(pi, e, tau, e)'
5000000 loops, best of 11: 96 nsec per loop

Accuracy tests. n is the number of dimensions followed by the weighted average number of correct bits (higher is better). The list contains counts for number of bits of error (lower counts are better). hypot12 is the baseline and hypot15 is the new code:

n=2 108.94  hypot12 [(104, 465), (105, 9650), (106, 74210), (107, 177328), (108, 220778)
n=2 109.12  hypot15 [(104, 76), (105, 6170), (106, 65987), (107, 163375), (108, 206841)

n=3 108.48  hypot12 [(103, 8), (104, 486), (105, 13420), (106, 93712), (107, 229769), (108, 243386)
n=3 108.72  hypot15 [(104, 102), (105, 7086), (106, 82426), (107, 208486), (108, 230416)

n=5 108.17  hypot12 [(103, 11), (104, 1353), (105, 18261), (106, 123382), (107, 253589), (108, 250362)
n=5 108.40  hypot15 [(104, 174), (105, 10427), (106, 108216), (107, 232246), (108, 243693)

n=10    107.97  hypot12 [(103, 33), (104, 3417), (105, 33754), (106, 150365), (107, 261133), (108, 237118)
n=10    108.18  hypot15 [(103, 5), (104, 653), (105, 24797), (106, 132075), (107, 243443), (108, 238205)

n=20    107.92  hypot12 [(102, 1), (103, 193), (104, 3811), (105, 39405), (106, 153847), (107, 263381), (108, 234462)
n=20    108.10  hypot15 [(103, 59), (104, 2030), (105, 29794), (106, 136819), (107, 247027), (108, 238805)

n=50    107.69  hypot12 [(103, 57), (104, 4987), (105, 53502), (106, 188830), (107, 275831), (108, 217911)
n=50    107.80  hypot15 [(103, 97), (104, 4406), (105, 47363), (106, 175900), (107, 265514), (108, 223802)

n=100   107.29  hypot12 [(103, 397), (104, 14872), (105, 101681), (106, 240449), (107, 264715), (108, 178790)
n=100   107.33  hypot15 [(103, 615), (104, 18598), (105, 100409), (106, 228175), (107, 258464), (108, 183213)

n=200   106.62  hypot12 [(102, 19), (103, 4065), (104, 55179), (105, 204706), (106, 282193), (107, 210798), (108, 119011)
n=200   106.60  hypot15 [(102, 31), (103, 5541), (104, 59552), (105, 205935), (106, 278179), (107, 208483), (108, 118478)

n=1000  104.68  hypot12 [(100, 102), (101, 5593), (102, 57169), (103, 184363), (104, 276279), (105, 218668), (106, 126077)
n=1000  104.64  hypot15 [(100, 119), (101, 5805), (102, 60339), (103, 193004), (104, 276688), (105, 213714), (106, 122384)

# Case for very large n but with fewer trials:

n=10000 100.49  hypot12 [(96, 62), (97, 1261), (98, 8502), (99, 20909), (100, 25938), (101, 19875), (102, 11594)
n=10000 100.48  hypot15 [(96, 61), (97, 1271), (98, 8530), (99, 20922), (100, 26032), (101, 19856), (102, 11562)

n=50000 97.72   hypot12 [(93, 5), (94, 521), (95, 5593), (96, 18338), (97, 26766), (98, 22070), (99, 12926)
n=50000 97.72   hypot15 [(93, 5), (94, 531), (95, 5582), (96, 18319), (97, 26805), (98, 22049), (99, 12970)

https://bugs.python.org/issue41513

[bedevere-bot]

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