About Accuracy and Precision
Throughout this section the following letters and names will be used:
- n
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the degree of linearization of exponentiation
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- precreq(n, x)
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the working-precision required to perform linearization
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- sigfig(n, x, y)
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the number of significant digits expected from tetration
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- offset(x)
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the ammount of offset, offset(x) = abs(x - round(x))
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The reason why I have not taken the time to find a
sigfig formula
for the super-logarithm is that it generally is much more "accurate"
than tetration. Tetration is also more "steep" than the super-logarithm,
so its harder to find "accurate" values with tetration. This is why
I have taken the time to find a
sigfig formula for tetration.
Working Precision
These are the formulas I have so far for working-precision. If the
value of working-precision p is below the output of this function,
then there is likely to be major round-off accumulation, and
the resulting solutions of the n-th degree linearization will
be zero. To prevent this, choose a working-precision
above the output of this function to perform an n-th degree
linearization.
First, the definition:
- precreq(n, x)
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:= while
(linearize(n, x, p)[2,2] != linearize(n, x)[2, 2]) { p++ }; return p;
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And here are some formulas:
- precreq(n, 2)
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=
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- precreq(n, e)
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= −6.7733 + 1.4314*n + 0.009*n^2
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- precreq(n, 10)
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=
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- precreq(n, x)
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=
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Arabic
Chinese
Latin
Significant Figures
I am still investigating the dynamics of this function, so I
don't have any final results, but here are two forms of the
function that I have some intermediate results about:
- sigfig(2, e, y)
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= offset(y)*(x^(y - 1)^2 - 1)/(7*y^3)
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- sigfig(n, e, π)
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= 0.231578 + 1.20731*log(n)
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Under Construction.
