IEEE 754 NUMBER SYSTEM
- 1. IEEe 754 number SYSTEM Paper Name : COMPUTER ARCHITECHTURE Paper code : EC502 Student Name : Debjyoti Musib Univ. Roll : 13000320104 Department : ECE MAKAUT CA1 EC502 1 27-07-2022
- 2. 27-07-2022 MAKAUT CA1 EC502 2 CONTENTS : INTRODUCTION COMPONENTS EXPLANATIONS EXAMPLES
- 3. 27-07-2022 MAKAUT CA1 EC502 3 What is IEEE 754 number system? The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point computation which was established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). IEEE in 1985 and augmented in 2008 provided a standard to represent floating-point numbers and process them. IEEE Standard 754 floating point is the most common representation today for real numbers on computers, including Intel-based PC’s, Macs, and most Unix platforms. There are several ways to represent floating point number but IEEE 754 is the most efficient in most cases.
- 4. 27-07-2022 MAKAUT CA1 EC502 4 IEEE 754 has 3 basic components: The Sign of Mantissa – This is as simple as the name. 0 represents a positive number while 1 represents a negative number. The Biased exponent – The exponent field needs to represent both positive and negative exponents. A bias is added to the actual exponent in order to get the stored exponent. The Normalised Mantissa – The mantissa is part of a number in scientific notation or a floating-point number, consisting of its significant digits. Here we have only 2 digits, i.e. O & 1. So a normalised mantissa is one with only one 1 to the left of the decimal.
- 5. 27-07-2022 MAKAUT CA1 EC502 5 IEEE 754 numbers are divided into two based on the above three components: Single precision. (32 bit) Double precision. (64 bit) Single-Precision - 1. Sign- In single precision, 1 bit is assigned for the sign (positive or negative). 2. Exponent- 8 bit is assigned for the range named exponent. 3. Mantissa- 23 bit is assigned for the precision ( it's for the fractional part)
- 6. 27-07-2022 MAKAUT CA1 EC502 6 Double-precision- 1. Sign- In single precision, 1 bit is assigned for the sign(positive or negative). 2. Exponent- 8 bit is assigned for the range named as an exponent. 3. Mantissa- 23 bit is assigned for the precision ( it's for the fractional part). Excess = 2n-1- 1 In single precision In double precision N= E = 8 N= E = 11 So, Excess = 28-1- 1 = 127 So, Excess = 211-1- 1 = 1023
- 7. 27-07-2022 MAKAUT CA1 EC502 7 Example: 85.125 85 = 1010101 0.125 = 001 85.125 = 1010101.001 =1.010101001 x 2^6 sign = 0 1. Single precision: biased exponent 127+6=133 133 = 10000101 Normalised mantisa = 010101001 we will add 0's to complete the 23 bits The IEEE 754 Single precision is: = 0 10000101 01010100100000000000000 This can be written in hexadecimal form 42AA4000
- 8. 27-07-2022 MAKAUT CA1 EC502 8 2. Double precision: biased exponent 1023+6=1029 1029 = 10000000101 Normalised mantisa = 010101001 we will add 0's to complete the 52 bits The IEEE 754 Double precision is: = 0 10000000101 0101010010000000000000000000000000000000000000000000 This can be written in hexadecimal form 4055480000000000
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