Machine epsilon, ε: What is step to next smaller value from 1?

Question

We know that machine epsilon, ε, is ”the difference between 1 and the least value greater than 1 that is representable in the given floating point type”.¹

However, the definition says nothing about the next smaller value. What is the difference between 1 and the next smaller value?

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Note

¹ ISO/IEC C 2011 standard, section 5.2.4.2.2, paragraph 13 (not authoritative on floating point but one example of the common description of machine epsilon).

Answer 1

No.

1-ε/2 is the largest number below 1. 1+ε is the smallest number above 1.

The significand for a normalized floating-point number is always in the range [1,2). Therefore, to represent numbers in the range [0.5,1), the exponent is -1 and the LSB is half as large.

So in binary, 1 + ε is something like:

1 + ε = 1.000...0001 ⨯ 2⁰

You can see that there's no number between 1 and 1 + ε, which matches the definition for ε.

But 1 - ε is something like:

1 - ε = 1.111...1110 ⨯ 2^-1

So there is exactly one number between 1 - ε and 1, that is, 1 - ε/2.