Floating Point Numbers - Machine numbers

Consider the set of machine numbers $$M(10, 2, 0)$$. (The "zero-length" for the exponent is to be understood such that there is only the sign ± and 0 available for the exponent. We interpret "+" as "+1" and "−" as "−1". The available exponential factors are thus $$10^{+1}, 10^0, 10^{−1}$$.)

Perform the addition $$\begin{equation} 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{12}\end{equation}$$ first from left to right $$\begin{equation} (...((1+\frac{1}{2})+\frac{1}{3})+...)+\frac{1}{12}\end{equation}$$ and then from right to left $$\begin{equation} 1+(...+(\frac{1}{10}+(\frac{1}{11}+\frac{1}{12}))...)\end{equation}$$ in the set $$M(10,2,0).$$

Start by first mapping the set of real numbers $$1,\frac{1}{2},\frac{1}{3},...,\frac{1}{12}\in \mathbb{R}$$ onto $$M(10,2,0)$$ via the "float operator" $$fl$$. For simplicity, we assume that $$fl$$ works by chopping off the digits for which there is not enough storage in $$M(10, 2, 0).$$

Which summation order gives the more accurate result, when compared to the results of the same calculation performed in $$\mathbb{R}$$?

So I'm studying ODEs and this exercise just came up. I can't figure out how to do it correctly. My idea is that we convert all the fractions into exponential notation, i.e. $$fl(\frac{1}{6})=1.66\cdot 10^{-1}$$. but I'm not sure how to compute the following:

first from left to right $$\begin{equation} (...((1+\frac{1}{2})+\frac{1}{3})+...)+\frac{1}{12}\end{equation}$$ and then from right to left $$\begin{equation} 1+(...+(\frac{1}{10}+(\frac{1}{11}+\frac{1}{12}))...)\end{equation}$$ in the set $$M(10,2,0).$$

Should I convert all the fractions and then add them up or should I do it like this:

$$\begin{equation} fl(fl(...fl(1+\frac{1}{2})...)) \end{equation}$$

EDIT: I just tried it again. Let's consider from right to left, then I have $$\begin{equation} fl(\frac{1}{11}+\frac{1}{12}) \end{equation}$$ Now I continue to $$\begin{equation} fl(fl(\frac{1}{11}+\frac{1}{12})+\frac{1}{10}) \end{equation}$$ And $$\begin{equation} fl(fl(fl(\frac{1}{11}+\frac{1}{12})+\frac{1}{10})+\frac{1}{9}) \end{equation}$$ If I keep going I end with 3.101

If I do from left to right by the same method, I get 3.1. Is it the correct way to do it?

• I think both. All the fractions themselves, and all the intermediate results are stored/computed by the machine so should be in the form that the machine handles. – Jaap Scherphuis Oct 3 '18 at 13:50

• Yes thank you. The problem is that I don't know how to convert it correct. I.e. $fl(1/3)=3.33\cdot 10^{-1}$ or $fl(1/3)=0.33\cdot 10^{0}$ how about $fl(1/7)=0.14\cdot 10^0$ can I do $fl(1/7)=1.42\cdot 10^{-1}$ – Joey Adams Oct 3 '18 at 15:37