We have that: $\epsilon$ is the smallest positive machine number that summed to $1$ resuts in $(\epsilon + 1)$: i.e. the smallest number greater than $1$:

if $p$ is precision and $\beta$ the base:

$$\begin{array}{lcl} 1 & = & 0.1\underbrace{00 \ldots 00}_{p-1 \text{ zeros}} \times \beta^1 \\ 1 + \epsilon & = & 0.1\underbrace{00 \ldots 0}_{p-2}1 \times \beta^1 \end{array}$$ we can consider $\epsilon$ as the distance between the above numbers: $$ \begin{array}{lcl} \epsilon & = & (1+\epsilon) - 1 \\ & = & 0.1\underbrace{00 \ldots 0}_{p-2}1 \times \beta^1 - 0.1\underbrace{00 \ldots 00}_{p-1} \times \beta^1 \\ & = & 0.\underbrace{000 \ldots 0}_{p-1}1 \times \beta^1 \\ & = & 1 \times \beta^{1} \times \beta^{-p} \\ & = & 1 \times \beta^{1-p} \\ & = & \beta^{1-p} \end{array} $$

so the only significant figure $1$, stays on the p-th position after the dot.
Calculating the value of that $p$, is equivalent to know the precision, i.e. the number of digits for the mantissa.

I know that the algorithm for finding Epsilon Machine is similar to this:

U = 1.0
while (1+U)>1 do
   Umem = U
   U = U/beta
end while
U = Umem

and I know that, from the given $\epsilon$, I can calculate the number $p$ of digits for the mantissa:

$$\begin{array}{lcl} \epsilon & = & \beta^{1-p} \\ \log\epsilon & = & \log\beta^{1-p} \\ \log\epsilon & = & (1-p)\log\beta \\ p & = & 1-\frac{\log \epsilon}{\log \beta} \end{array} $$

I don't know how elaborate this fact in Python, can you give me any hint? Thanks.


1 Answer 1


It's easier to look these things up than to try and write a program that tests these quantities. Floating point numbers used by almost all programming languages today are specified in the IEEE-754 standard, see https://en.wikipedia.org/wiki/IEEE_754

For floating point numbers, Python uses the "double precision" (binary64) format, for which you can find the specifications in the table about a page down on the page referenced. If you click on "binary64" there, you get to a page solely on double precision: https://en.wikipedia.org/wiki/Double-precision_floating-point_format


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