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1.10: Machine Epsilon

Last updated

May 31, 2022
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- 1.9: Inexact Numbers
- 1.11: IEEE Double Precision Format

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Machine epsilon $\left(\epsilon_{\mathrm{mach}}\right)$ is the distance between 1 and the next largest number. If $0 \leq \delta<\epsilon_{\mathrm{mach}} / 2$ , then $1+\delta=1$ in computer math. Also since

$x+y=x(1+y / x) \nonumber$

if $0 \leq y / x<\epsilon_{\operatorname{mach}} / 2$ , then $x+y=x$ in computer math.

Find $\epsilon_{\text {mach }}$

The number 1 in the IEEE format is written as

$1=2^{0} \times 1.000 \ldots 0, \nonumber$

with $230^{\prime}$ s following the binary point. The number just larger than 1 has a 1 in the 23 rd position after the decimal point. Therefore,

$\epsilon_{\text {mach }}=2^{-23} \approx 1.192 \times 10^{-7} \nonumber$

What is the distance between 1 and the number just smaller than 1? Here, the number just smaller than one can be written as

$2^{-1} \times 1.111 \ldots 1=2^{-1}\left(1+\left(1-2^{-23}\right)\right)=1-2^{-24} \nonumber$

Therefore, this distance is $2^{-24}=\epsilon_{\operatorname{mach}} / 2$ .

The spacing between numbers is uniform between powers of 2, with logarithmic spacing of the powers of 2 . That is, the spacing of numbers between 1 and 2 is $2^{-23}$ , between 2 and 4 is $2^{-22}$ , between 4 and 8 is $2^{-21}$ , etc. This spacing changes for denormal numbers, where the spacing is uniform all the way down to zero.

Find the machine number just greater than 5

A rough estimate would be $5\left(1+\epsilon_{\text {mach }}\right)=5+5 \epsilon_{\text {mach }}$ , but this is not exact. The exact answer can be found by writing

$5=2^{2}\left(1+\frac{1}{4}\right) \nonumber$

so that the next largest number is

$2^{2}\left(1+\frac{1}{4}+2^{-23}\right)=5+2^{-21}=5+4 \epsilon_{\text {mach }} \nonumber$

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