Numerical Analysis Assignment #1 — IEEE Arithmetic  
Question #1
Determine the maximum and minimum number that can be stored in:
(a)  A 5-byte unsigned integer
(b)  A 5-byte signed integer
08.20.16
Question #2
Consider a real number stored with 5 bytes. Bit #1 is reserved for the sign, while bits #2 to #10 are reserved for the biased exponent, and bits #11 to #40 are related to the significand. Do the following:
(a)  Find the minimum and maximum possible exponent $p$
(b)  Find the smallest possible positive number
(c)  Find the largest possible number
(d)  Find the smallest possible positive subnormal number
08.29.16
Question #3
Consider a real number stored with 6 bytes. Bit #1 is reserved for the sign, while bits #2 to #13 are reserved for the biased exponent, and bits #14 to #48 are related to the significand. Find the machine precision $\epsilon_{\rm mach}$.
Question #4
Say that $$ x=-g + \sqrt{g^2+1} $$ Say that both $x$ and $g$ are stored in memory using single precision numbers with a relative error due to machine accuracy of $\epsilon_{\rm mach}=2\times 10^{-10}$. Do the following:
(a)  Find the relative error on $x$ given $g=1000.0$
(b)  Recast the equation for $x$ in difference form to reduce its relative error
(c)  Find the relative error on $x$ given $g=1000.0$ for the recast equation outlined in (b)
08.30.16
Question #5
It is desired to minimize the number of bits that can store a certain range of numbers. The range lower limit is $3\times 10^{-65}$, and the range upper limit is $10^{32}$. Do the following:
(a)  Find the minimum number of bits needed to store the exponent.
(b)  Find the minimum number of bits needed to store the significand.
(c)  Find the total number of bits needed.
09.13.17
Answers
2.  255, $-254$, $3.454\times 10^{-77}$, $1.15792089 \times 10^{77}$, $3.217\times 10^{-86}$.
Due on Wednesday September 20th at 16:30. Do Questions #1, #2, #3, and #5 only.
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