Comments (10)
KlausC
commented on July 25, 2024
Further investigation shows, that the powers of big integers are unreliable for ~ n < -2^53:
The first column uses Int, the second Float64 exponent type. Third column the correct value.
julia> [45:62 prevfloat(1.0).^(-2 .^ (45:62)) prevfloat(1.0).^float(-2 .^ (45:62)) exp.(log(big(prevfloat(1.0))) .* (-2 .^ (45:62)))]
18×4 Matrix{BigFloat}:
45.0 1.00391 1.00391 1.00391
46.0 1.00781 1.00781 1.00784
47.0 1.01562 1.01562 1.01575
48.0 1.03123 1.03123 1.03174
49.0 1.06233 1.06233 1.06449
50.0 1.12352 1.12352 1.13315
51.0 1.23654 1.23654 1.28403
52.0 1.35914 1.35914 1.64872
53.0 1.10235e-07 1.10235e-07 2.71828
54.0 -54.5981 -54.5981 7.38906
55.0 -8942.87 -8942.87 54.5982
56.0 -6.22028e+07 -6.22028e+07 2980.96
57.0 -1.18444e+15 -1.18444e+15 8.88611e+06
58.0 -1.9329e+29 -1.9329e+29 7.8963e+13
59.0 -2.44925e+57 -2.44925e+57 6.23515e+27
60.0 -1.91951e+113 -1.91951e+113 3.88771e+55
61.0 -5.82523e+224 -5.82523e+224 1.51143e+111
62.0 Inf Inf 2.28441e+222
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mbauman
commented on July 25, 2024
The issue here is that inv(x) returns a rounded Float64 of the inverse — with 52 bits precision. In fact, we start losing 1ulp accuracy at -2^27 and it just goes downhill from there. Seems we need a better strategy here.
Line 1322 in e07c0f1
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oscardssmith
commented on July 25, 2024
@mbauman that gets compensated for on
Line 1324 in e07c0f1
I think something must be going wrong in the error compensation though. Investigating now.
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mbauman
commented on July 25, 2024
Ah, I see. We still are missing compensation somewhere here:
julia> ulps(b, x) = abs(b^x - Float64(big(b)^x))/eps(b^x)
ulps (generic function with 3 methods)
julia> [ulps(prevfloat(1.0), -2^e) for e in 26:54]
29-element Vector{Float64}:
0.0
1.0
2.0
8.0
32.0
128.0
512.0
2048.0
8192.0
32768.0
131073.0
524302.0
2.097258e6
8.389461e6
3.3561258e7
1.34272348e8
5.37307984e8
2.15098174e9
8.617942835e9
3.4584177964e10
1.3924045658e11
5.6426422088e11
2.316653480914e12
9.762831672521e12
4.3352610026219e13
2.13851005933614e14
1.304153939836051e15
2.0538755963422595e23
8.723923242651639e15from julia.
KlausC
commented on July 25, 2024
When I fixed the first issue (typemin(Int)), I found the second (inaccuracy) disappeared. Will drop PR soon.
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oscardssmith
commented on July 25, 2024
@KlausC that suggests something you were doing is wrong. From what I can tell, this appears to be an issue caused by me skipping a term in the error compensation that is very small but can accumulate to a significant amount. At this point, I believe that I have corrected for the case of power of 2 exponents, but not for the case of exponents with bits set.
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KlausC
commented on July 25, 2024
that suggests something you were doing is wrong. From what I can tell, this appears to be an issue caused by
That is possible. I fixed the first issue primarily. As a side-effect the exploding errors for prevfloat(1.0) powers disappeared. Of course it is possible, that there are more improvements needed.
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KlausC
commented on July 25, 2024
BTW, I repeated the ULPs counting after my fix:
julia> ulps(b, x) = abs(b^x - Float64(big(b)^x))/eps(b^x)
ulps (generic function with 1 method)
julia> [26:63 [ulps(prevfloat(1.0), -2^e) for e in 26:63]]
38×2 Matrix{Float64}:
26.0 0.0
27.0 0.0
28.0 0.0
29.0 0.0
30.0 0.0
31.0 0.0
32.0 0.0
33.0 0.0
34.0 0.0
35.0 0.0
36.0 0.0
37.0 0.0
38.0 0.0
39.0 0.0
40.0 0.0
41.0 0.0
42.0 0.0
43.0 0.0
44.0 0.0
45.0 0.0
46.0 0.0
47.0 0.0
48.0 0.0
49.0 0.0
50.0 0.0
51.0 0.0
52.0 0.0
53.0 1.0
54.0 1.0
55.0 3.0
56.0 12.0
57.0 34.0
58.0 144.0
59.0 646.0
60.0 3255.0
61.0 10322.0
62.0 51893.0
63.0 NaN
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oscardssmith
commented on July 25, 2024
what about [26:63 [ulps(prevfloat(1.0), 1-2^e) for e in 26:63]]?
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KlausC
commented on July 25, 2024
Also fine:
julia> [26:63 [ulps(prevfloat(1.0), -2^e+1) for e in 26:63]]
38×2 Matrix{Float64}:
26.0 0.0
27.0 1.0
28.0 0.0
29.0 0.0
30.0 0.0
31.0 0.0
32.0 0.0
33.0 0.0
34.0 0.0
35.0 0.0
36.0 0.0
37.0 1.0
38.0 0.0
39.0 1.0
40.0 0.0
41.0 0.0
42.0 0.0
43.0 0.0
44.0 0.0
45.0 0.0
46.0 1.0
47.0 0.0
48.0 1.0
49.0 1.0
50.0 1.0
51.0 0.0
52.0 1.0
53.0 0.0
54.0 0.0
55.0 2.0
56.0 11.0
57.0 33.0
58.0 144.0
59.0 646.0
60.0 3254.0
61.0 10321.0
62.0 51892.0
63.0 NaN
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