Methods and systems of computer arithmetic

Author: Janusz Biernat

ISBN: 83-7085-590-3
Issue: 2, 2001.
Format: B5, p. 170
Binding: soft
Publisher: PWR

About the book

The book contains an explanation of basic arithmetic algorithms used in modern computers. This includes the basic actions of fixed-point and floating-point arithmetic, such as: addition, subtraction, multiplication, division and calculation of the square root, as well as methods for calculating the values of elementary functions (trigonometric, logarithmic and exponential). For individual activities, the methods of controlling the correctness of results and ensuring the required accuracy of calculations are discussed.
Formal descriptions of the discussed methods and algorithms are illustrated by arrangement solutions in the form of logical structures. Methods of accelerating arithmetic operations are also discussed. Standardized characteristics of their complexity and speed were given for the presented systems.
The appendix at the end of the appendix gives a brief description of the IEEE 754/854 floating point standard.
The book is intended for students of computer science, engineers dealing with the practical implementation of the principles of computer arithmetic and for all who would like to know the answers to the questions: how does the computer counts and why it does so quickly?

Table of Contents

Admission

1. From scratch to the computer

2. Number representation

2.1. Continuous systems (with fixed basis)
2.2. Representation of negative numbers in standby systems
2.3. Natural systems and base conversions
2.3.1. Conversion of the base in natural systems
2.3.2. Conversion to the associated database in natural systems
2.3.3. Base conversion in binary systems
2.3.4. Pseudonatural representations of negative numbers
2.4. Supplementary systems
2.4.1. Binary replacement systems
2.4.2. Binary quasi-symmetrical polarized representations
2.4.3. Relationship between supplementary representation and representation in the SD system
2.5. Rest systems
2.5.1. Conversion from a fixed-time system to a residual system
2.5.2. Conversion from a residual system to a continuous system
2.5.3. Square-residual QRNS system
2.6. Floating-point representations
2.6.1. Unambiguity of floating-point representation
2.6.2. Accuracy and range of floating point representation
2.7. Logarithmic representation of numbers

3. Basic arithmetic operations

3.1. Addition and subtraction
3.1.1. Addition and subtraction in natural positional systems
3.1.2. Adding and subtracting the character-module in the system
3.1.3. Addition and subtraction in a system with a marked digit
3.1.4. Addition and subtraction in supplementary systems
3.1.5. Adding multiple arguments
3.1.6. Addition and subtraction in the rest system
3.2. Multiplication
3.2.1. Multiplication in a continuous system
3.2.2. Multiplication in replacement systems
3.3. Division
3.4. Multiplication and sequential division in residual systems
3.5. Calculation of the square root
3.6. Floating point operations
3.6.1. Rounding the result
3.6.2. Protecting digit and rounding numbers
3.6.3. Cumulation of errors while performing arithmetic operations

4. Logic of digital circuits

4.1. Logic algebra and logic functions
4.2. Structural elements of digital circuits
4.2.1. Simple logic gates (funktory)
4.2.2. Remembering elements - flip-flops and registers
4.2.3. Evaluation of the complexity of digital circuits
4.3. Digital arithmetic systems
4.3.1. 1-bit adder - AT structures and characteristics
4.3.2. Other elementary arithmetic systems - AT structures and characteristics
4.4. Limits of the speed of arithmetic operations

5. Arithmetic operations in binary systems

5.1. Addition and subtraction in binary systems
5.1,1. Addition and subtraction in a natural binary system
5.1.2. Addition and subtraction in binary replacement systems
5.1.3. Adding and subtracting the character-module in the system
5.1.4. Addition and subtraction in binary codes with a load
5.1.5. Addition and subtraction in a system with a marked digit
5.1.6. Multi-operand addition in a natural binary system
5.1.7. Addition and subtraction of binary in the residual RNS system
5.1.8. Addition and subtraction with saturation
5.2. graduation
5.3. Sequential multiplication algorithms

5.3.1. Multiplication in a continuous system
5.3.2. A sequence multiplication of numbers with a sign in binary systems
5.4. Sequential division
5.4.1. Sharing (restitution)
5.4.2. Dividing non-performing (unrestrictive)
5.5. Calculation of the square root

6. Fast adders

6.1. Combiner with anticipation of transfers
6.2. Combiner with transfer jumps
6.3. Conditional adders
6.4. Multi-column combiners

7. Methods of Acceleration of multiplication

7.1. Reduction in the number of partial products
7.1.1. Booth's algorithm
7.1.2. Booth extended algorithm
7.1.3. Implementation of the Booth algorithm
7.1.4. Other modifications of the Booth algorithm
7.1.5. Recoding the multiplier in the top-up system to 1
7.2. Structuring of multipliers
7.2.1. Modular design of multipliers
7.2.2. Accumulation of partial products
7.3. Matrix multiplication systems

8. Fast division methods

8.1. Simplification and acceleration of comparison
8.2. Splitting in an enhanced basis
8.3. Matrix dividing systems
8.4. Fast algorithm for calculating the square root

9. Computational methods of computational arithmetic

9.1. Numeric division
9.1.1. Bringing the rational quotient to its finite development
9.1.2. Splitting by dividing the divisor
9.2. Methods for calculating the values of elementary functions
9.3. Procedures of additive and multiplication normalization
9.3.1. The exponential function
9.3.2. Logarithmic function
9.3.3. Trigonometric functions
9.3.4. Inverse trigonometric functions
9.3.5. Hyperbolic functions
9.4. Vector rotation method in a coordinate system (CORDIC)

10. The arithmetic and logic unit

10.1. Fixed-point unit
10.2. Floating point arithmometer

D1. IEEE 754/854 floating point standards
D1.1. Representation of numbers in the IEEE 754 standard
D1.2. Exceptions, their handling, rounding modes

Literature
Polish-English dictionary of more important dates