CLD-Lec9-Fixed-Floating-Point-Numbers-Extended

Lecture Overview

  • Topic: Fixed and Floating Point Numbers

  • Instructor: Dr. Ngoc-Son Pham

  • Course: COSE221 Digital Logic Design

  • Institution: Computer Science & Engineering, Korea University

C Code Examples

Example 1

#include <stdio.h>
int main() {
   signed int sa = 7;
   signed int sb = -7;
   unsigned int ua = *((unsigned int *) &sa);
   unsigned int ub = *((unsigned int *) &sb);
   printf("sa = %d : ua = 0x%x\n", sa, ua);
   printf("sb = %d : ub = 0x%x\n", sb, ub);
   return 0;
}

  • Output Interpretation:

    • Displaying signed and unsigned integer representations.

Example 2

#include <stdio.h>
int main() {
   float f1 = -58.0;
   unsigned int u1 = *((unsigned int *) &f1);
   printf("f1 = %f\n", f1);
   printf("f1 = %3.20f\n", f1);
   printf("u1 = 0x%X\n", u1);
   return 0;
}

  • Understanding Floating-Point Representation:

    • Shows conversion from float to unsigned integer.

Example 3

#include <stdio.h>
int main() {
   double d1 = -58.0;
   unsigned long long u1 = *((unsigned long long *) &d1);
   printf("d1 = %lf\n", d1);
   printf("d1 = %3.20lf\n", d1);
   printf("u1 = 0x%llX\n", u1);
   return 0;
}

  • Output Interpretation:

    • Display double value and its unsigned long long representation.

Fixed-Point and Floating-Point Representation

Fixed-Point Numbers

  • Definition: Implies a binary point fixed at a certain position.

    • Example:

    • Representation of 6.75 with 4 integer bits and 4 fraction bits: 0110.1100

  • Agreement Required: Fixed-point representation needs agreement on number of integer/fraction bits.

Signed Fixed-Point Numbers

  • Representation: Can be in sign/magnitude or two’s complement.

  • Example: -2.375:

    • Sign/magnitude: 1010 0110

    • Two’s complement:

      1. Flip bits: 1101 1001

      2. Add 1: 1101 1010

Example Calculation

  • Calculation of 0.75 + (-0.625):

    • 0.75 = 0000 1100

    • -0.625 = 1111 0110

Floating-Point Numbers

  • Purpose: Allow representation of very large/small numbers, binary point floats

  • Scientific Notation Format: ± M × B^E

  • Structure for IEEE 754 32-bit representation:

    • 1 sign bit, 8 exponent bits, 23 mantissa bits

Floating-Point Representation Steps

  1. Convert decimal to binary.

  2. Assign sign bit, exponent, mantissa.

  3. Use biased exponent - IEEE 754 standard uses 127.

Special Cases in Floating-Point Numbers

  • IEEE 754 Standard: Special Cases

    • Zero, ∞, NaN (Not a Number)

Floating-Point Precision

  • Single Precision:

    • 32-bit (1 sign, 8 exponent, 23 fraction)

    • Range: ±1.175494 x 10^−38 to ±3.402824 x 10^38

  • Double Precision:

    • 64-bit (1 sign, 11 exponent, 52 fraction)

    • Range: ±2.22507385850720 x 10^−308 to ±1.79769313486232 x 10^308

Rounding in Floating-Point Operations

  • Rounding modes:

    • Round down, up, toward zero, round to nearest.

Floating-Point Addition Steps

  1. Extract exponent/fraction bits.

  2. Prepend leading 1, compare exponents.

  3. Shift as needed, add mantissas.

  4. Normalize and round the result.

Floating-Point Multiplication Process

  1. Multiply Signs:

  • XOR to determine the sign.

  1. Add Exponents:

  • Adjust according to bias.

  1. Multiply Mantissas:

  • Normalize result following the multiplication.

Example Calculation: Multiply 3.0 x 2.5

  • Convert to IEEE 754 format and perform calculation.

Binary Coded Decimal (BCD)

  • Concept: Encoding decimal digits in binary.

  • Example BCD representation: 1.7 = 0001 . 0111