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If the distance between 7 and the next larger floating-point number is 2^-12. What is machine epsilon on that computer? What is the distance between 70 and the next larger floating-point number on that com- puter? Assume of course that the computer represents numbers in base 2.

-Should the machine epsilon just be 2^-12? -since machine epsilon is the smallest floating point between two number, so does it change between 70 and next number?

M Z
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  • With my version of Matlab, and with my computer, asking > eps, I get 2.2204e-16. – Jean Marie Jan 22 '19 at 22:03
  • This epsilon can be met in very different circumstances even when it is qualified as "noise" ; see for example my answer to this question : https://math.stackexchange.com/q/2246240 – Jean Marie Jan 22 '19 at 23:09

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Machine precision epsilon is usually the smallest epsilon that we can add to 1 such that it is distinct.

7 is encoded as $.111\times 2^3$ while 1 is encoded as $.1\times 2^1$.

Since $2^{-15}\times 2^3=2^{-12}$ we must have that $\epsilon=2^{-15}\times 2^1=2^{-14}$. It also means that we have a mantissa of 15 bits.

Klaas van Aarsen
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  • thanks for the helping, I'm still confused about where the 2^3 comes from, and the 2^-15 by 2^-12/2^3, how do they have these relations ? – M Z Jan 22 '19 at 04:43
  • 2^3 is the smallest power of 2 that is larger than 7, which we need since the mantissa must be less than 1. And 2^-15 follows from that power combined with the given 2^-12. – Klaas van Aarsen Jan 22 '19 at 04:59