FLOATING-POINT NUMBER REPRESENTATION Dr. Konstantinos Tatas ACOE161 - Digital Logic for Computers - Frederick University

The range/accuracy problem The range of numbers that can be represented with n bits is In 2’s complement: from - /2 to /2 -1 For n=8: From –128 to +127 For n=16: From –32,768 to +32,767 Still, in many application an even larger range is required ACOE161 - Digital Logic for Computers - Frederick University

ACOE161 - Digital Logic for Computers - Frederick University Real numbers Instead of representing the actual value, in the base system, we represent the sign, M, b and e ACOE161 - Digital Logic for Computers - Frederick University

FLOATING-POINT REPRESENTATION IEEE short real: 8 bits for the exponent (in Ex-127), 23 bits for the mantissa IEEE long real: 11 bits for the exponent, 52 bits for the mantissa Sign (S) Biased exponent (E) Unsigned normalized mantissa (M) ACOE161 - Digital Logic for Computers - Frederick University

ACOE161 - Digital Logic for Computers - Frederick University RESERVED VALUES ACOE161 - Digital Logic for Computers - Frederick University

Examples (IEEE short real format) Binary value Normalized Binary value exponent Biased Exponent (Excess -127 Sign, exponent, mantissa -1.01 127 1 01111111 0100000000000000000000 +1011.0101 +1.0110101 +3 130 0 01000010 0110101000000000000000 -0.0000011 -1.1 -6 121 1 01111001 1000000000000000000000 +11010101 +1.11010101 +7 134 0 10000110 1101010100000000000000 ACOE161 - Digital Logic for Computers - Frederick University

ACOE161 - Digital Logic for Computers - Frederick University Homework Convert the following 2’s complement values to IEEE short real floating-point representation 10011010 0110.0101 0.1111110 1100.0001 ACOE161 - Digital Logic for Computers - Frederick University