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?

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).

No, machine epsilon is not defined as “the smallest positive number st (1+ε)>1”. That definition defines a different number from what is intended, which is “the difference between the successor of 1 and 1”: http://blog.frama-c.com/index.php?post/2013/05/09/FLT_EPSILON — Pascal Cuoq, Sep 20 '13 at 06:27

Dietrich Epp · Accepted Answer · 2013-09-20T05:44:58.423

7

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.

edited Sep 20 '13 at 05:44

answered Sep 20 '13 at 05:34

Dietrich Epp

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How about `(1-ε/3)` isn't that larger than `(1-ε/2)` ? – Emond Sep 20 '13 at 05:35
Would you mind to point me to a reading about them, please? I want to get an idea of why this is the case. – Sean Sep 20 '13 at 05:37
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@ErnodeWeerd: 1-ε/3 = 1-ε/2, they are same floating point number. – Dietrich Epp Sep 20 '13 at 05:37
So (1-ε)<1 does hold. But (1-ε/2)==1 does NOT hold since (1-ε/2) is smaller than 1? – Sean Sep 20 '13 at 05:40
@Sean: Right, 1-ε/2<1. – Dietrich Epp Sep 20 '13 at 05:41
Example in Python: `>>> 1.0 - sys.float_info.epsilon / 2 0.9999999999999999 >>> (1.0 - sys.float_info.epsilon / 2) == (1.0 -sys.float_info.epsilon / 3) True` EDIT: bah, formatting. – Electro Sep 20 '13 at 05:42

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

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