
Real values which cannot be represented exactly in a floating point representation must be rounded to the nearest floating point value. The results of a floating point operation will in general be such values and thus there is a rounding error possible in any floating point operation. Referring to Figure \ref{floats.pdf} it can be seen that the largest possible rounding error is half the distance between successive floats; this means that rounding errors increase as the value to be represented increases. For the result of a particular operation, the maximum possible rounding error can be determined and is commonly expressed in ``units in the last place'' (ulp), with 1 ulp equivelant to half the distance between successive floats\cite{goldberg1991whatevery}.



{\bf Put this stuff in an Appendix?}
\subsection{Addition and Subtraction}

According to the IEEE-754 standard, if $e_1 < e_2$, then the preferred form of $f_1 + f_2$ is:
\begin{align}
	m_1 \beta^{e_1} \pm m_2 \beta^{e_2} &= (m_1 \pm \beta^{e_2 - e_1} m_2) \beta^{e_1}
\end{align}

This is equivelant to shifting the fixed point in $m_2$ by $e_2 - e_1$ to the left, and then performing fixed point addition or subtraction. If the result of the addition/subtraction requires a carry/borrow, divide result by $\beta$ (ie: shift digits by $1$ the right) and increment/decrement exponent. Then normalise the result (subtract leading zeros in mantissa from the exponent). Lastly perform the rounding operation; if this would generate a carry/borrow, shift right and increment/decrement exponent again, repeat.


\subsection{Multiplication and Division}
\begin{align}
	m_1 \beta^{e_1} \times m_2 \beta^{e_2} &= (m_1 \times m_2 ) \beta^{e_1 + e_2}
\end{align}

\begin{align}
	m_1 \beta^{e_1} \div m_2 \beta^{e_2} &= (m_1 \div m_2 ) \beta^{e_1 - e_2}
\end{align}

Multiplication and Division are not inverses.

Floating point operations can in principle be performed using integer operations, but specialised Floating Point Units (FPUs) are an almost universal component of modern processors\cite{kelley1997acmos}. The improvement of FPUs remains highly active in several areas including: efficiency\cite{seidel2001onthe}; accuracy of operations\cite{dieter2007lowcost}; and even the adaptation of algorithms originally used in software, such as Kahan's Fast2Sum algorithm\cite{kadric2013accurate}. 
