# What is the maximum number of significant bits lost when the computer evaluates x − y using IEEE 64 bits?

Consider two positive numbers $x = p2^m$ and $y = q2^n$ such that $m > n$, $1 < p < 2$, and $1 < q < 2$. Both of these numbers can be stored using the IEEE 64 bit standard.

What is the maximum number of significant bits lost when the computer evaluates $x − y$?

• This looks like a homework question to me. Hint: What happens during subtraction when $x$, $y$ are consecutive floating-point numbers? A related issue possibly of interest: Sterbenz lemma. – njuffa Mar 22 '17 at 18:54
• I am aware of this fact. This has to do something with "Loss of Precision Theorem". But I am unable to figure out the exact bit loss.. Can bit loss be greater that 11(i.e bits to save exponent )? – user427820 Mar 23 '17 at 12:46
• My suggestion: work through a subtraction of two consecutive floating-point numbers by hand. Note that the restrictions in the question do not seem to allow consecutive floating-point numbers (but two numbers that are very close). Considering the case of consecutive floating-point numbers first should be helpful in understanding the mechanics and should get you 95% of the way to the final answer. – njuffa Mar 24 '17 at 2:15