U. Glaeser

Earl E. Swartzlander, Jr.
University of Texas at Austin
Gensuke Goto
Yamagata University
9.1 High-Speed Computer Arithmetic
Earl E. Swartzlander, Jr.
Introduction
© 2002 by CRC Press LLC
9
Computer Arithmetic
9.1 High-Speed Computer Arithmetic
Introduction • Fixed-Point Number Systems • Fixed-Point
Arithmetic Algorithms • Floating Point
Arithmetic • Conclusions
9.2 Fast Adders and Multipliers
Introduction • Adder • Carry Skip
Adder • Multiplier • Conclusion
The speed of a computer is determined to a first order by the speed of the arithmetic unit and the speed
of the memory. Although the speed of both units depends directly on the implementation technology,
arithmetic unit speed also depends strongly on the logic design. Even for an integer adder, speed can
easily vary by an order of magnitude while the complexity varies by less than 50%.
This chapter section begins with a brief discussion of binary fixed point number systems in the subsection
on “Fixed Point Number Systems.” The subsection on “Fixed Point Arithmetic Algorithms” provides
examples of fixed point implementations of the four basic arithmetic operations (i.e., add, subtract,
multiply, and divide). Finally, the “Floating Point Arithmetic” subsection describes algorithms for floating
point arithmetic.
Regarding notation, capital letters represent digital numbers (i.e., n-bit words), while subscripted lower
case letters represent bits of the corresponding word. The subscripts range from 0 to n − 1 to indicate the bit
position within the word (x 0 is the least significant bit of X, x n−1 is the most significant bit of X, etc.). The
logic designs presented in this chapter are based on positive logic with AND, OR, and INVERT operations.
Depending on the technology used for implementation, different logical operations (such as NAND and
NOR) or direct transistor realizations may be used, but the basic concepts do not change significantly.
Fixed-Point Number Systems
Most arithmetic is performed with fixed-point numbers, which have constant scaling (i.e., the position
of the binary point is fixed). The numbers can be interpreted as fractions, integers, or mixed numbers,
depending on the application. Pairs of fixed-point numbers are used to create floating-point numbers,
as discussed in “Floating-Point Arithmetic” subsection.
At the present time, fixed-point binary numbers are generally represented using the two’s complement
number system. This choice has prevailed over the sign magnitude and one’s complement number systems,
because the frequently performed operations of addition and subtraction are easiest to perform on two’s