Computer Arithmetic in Practice

Chapter 5 Basic Arithmetic Operations on Floating-point Numbers

DOI: 10.1201/9781003363286-5

5.1 Addition

The purpose of the following discussion is to show how to perform the operation $\mathrm{Z}=\mathrm{X}+\mathrm{Y}$, where (5.1):

${\mathrm{M}}_{\mathrm{Z}}\cdot {\mathrm{p}}^{{\mathrm{E}}_{\mathrm{Z}}}={\mathrm{M}}_{\mathrm{X}}\cdot {\mathrm{p}}^{{\mathrm{E}}_{\mathrm{X}}}+{\mathrm{M}}_{\mathrm{Y}}\cdot {\mathrm{p}}^{{\mathrm{E}}_{\mathrm{Y}}}$

${\mathrm{M}}_{\mathrm{Z}}={\mathrm{M}\prime }_{\mathrm{X}}+{\mathrm{M}\prime }_{\mathrm{Y}}{\mathrm{E}}_{\mathrm{Z}}=\max \left({\mathrm{E}}_{\mathrm{X}}{,\mathrm{E}}_{\mathrm{Y}}\right)$ (5.1)

and (5.2)

$\begin{array}{cccc}{\mathrm{M}\prime }_{\mathrm{X}}={\mathrm{M}}_{\mathrm{X}},& {\mathrm{M}\prime }_{\mathrm{Y}}={\mathrm{M}}_{\mathrm{Y}}& \mathrm{for}& {\mathrm{E}}_{\mathrm{X}}={\mathrm{E}}_{\mathrm{Y}}\\ {\mathrm{M}\prime }_{\mathrm{X}}={\mathrm{M}}_{\mathrm{X}},& {\mathrm{M}\prime }_{\mathrm{Y}}={\mathrm{M}}_{\mathrm{Y}}\cdot {\mathrm{p}}^{-\left|{\mathrm{C}}_{\mathrm{X}}-{\mathrm{C}}_{\mathrm{Y}}\right|}& \mathrm{for}& {\mathrm{E}}_{\mathrm{X}}>{\mathrm{E}}_{\mathrm{Y}}\\ {\mathrm{M}\prime }_{\mathrm{X}}={\mathrm{M}}_{\mathrm{X}}\cdot {\mathrm{p}}^{-\left|{\mathrm{C}}_{\mathrm{X}}-{\mathrm{C}}_{\mathrm{Y}}\right|},& {\mathrm{M}\prime }_{\mathrm{Y}}={\mathrm{M}}_{\mathrm{Y}}& \mathrm{for}& {\mathrm{E}}_{\mathrm{X}}<{\mathrm{E}}_{\mathrm{Y}}\end{array}$ (5.2)

Addition of two mantises should be done according to the rules specified in Chapter 3. Mantises in 2's format are added according to the formula ${\mathrm{M}}_{\mathrm{Z}}={\mathrm{M}\prime }_{\mathrm{X}}+{\mathrm{M}\prime }_{\mathrm{Y}}$. Adding the mantises in SM format requires applying the rules given in Table 3.1. Sometimes the result of the addition cannot be stored on the assumed number of bits, so it should be normalized according to the following rules: