Adding floating-point numbers involves several steps due to the specific format these numbers are stored in memory. To understand this process, we'll explain the flow of adding two floating-point numbers using a flowchart-style explanation.

Flowchart for Adding Floating-Point Numbers

Here is a step-by-step guide to create the flowchart for adding two floating-point numbers:

1. Start: Begin the process.

2. Extract Components:

   • Floating-point numbers are typically represented in the format:
     $(-1)^s \times (1.M) \times 2^E$ where:
     • $s$ is the sign bit (0 for positive, 1 for negative),
     • $M$ is the mantissa (or significand),
     • $E$ is the exponent (adjusted by a bias).
   • Extract the sign bit, exponent, and mantissa from both numbers.

3. Align Exponents:

   • If the exponents of the two numbers are different, shift the mantissa of the number with the smaller exponent to the right until the exponents match.
   • Increment the exponent of the shifted mantissa accordingly.
   • Check: If the difference is larger than a certain limit (determined by precision), the smaller number may effectively be treated as zero due to underflow.

4. Add/Subtract Mantissas:

   • If the sign bits are the same, add the mantissas.
   • If the sign bits differ, subtract the smaller mantissa from the larger one (using the absolute value). The result will inherit the sign of the larger mantissa.

5. Normalize the Result:

   • After addition or subtraction, the result may need to be normalized:
     • Shift Left: If there are leading zeros in the mantissa, shift the mantissa left and decrement the exponent until the first digit of the mantissa is 1.
     • Shift Right: If the mantissa has overflowed (more bits than the format allows), shift right and increment the exponent.
   • Check for Overflow/Underflow:
     • If the exponent exceeds the maximum limit, it results in overflow.
     • If the exponent drops below the minimum limit, it results in underflow.

6. Round the Result:

   • Due to limited precision, round the mantissa according to the floating-point format’s rounding rules (e.g., round to nearest, round toward zero, etc.).

7. Pack the Result:

   • Reconstruct the floating-point number from the computed sign, exponent, and mantissa.

8. End: Finish the process with the final floating-point result.

Explanation of Each Step:

• Extract Components: Separating the parts of each floating-point number ensures we can perform bitwise operations separately on each component.

• Align Exponents: Aligning exponents by shifting mantissas ensures the numbers are in the same "place value," which is crucial for accurate addition or subtraction.

• Add/Subtract Mantissas: The mantissas are added or subtracted based on the sign bits, which allows the correct combination of values, whether they are of the same or opposite signs.

• Normalize the Result: Normalizing ensures that the floating-point number remains in the standard form, maintaining precision and avoiding overflow or underflow.

• Round the Result: Rounding the result helps accommodate the precision limits of floating-point formats, ensuring that the result is as accurate as possible given the constraints.

• Pack the Result: Combining the components back into a single floating-point number allows it to be used in subsequent calculations.

Visual Flowchart Representation:

To visualize this as a flowchart:

1. Start →
2. Extract Components →
3. Align Exponents →
4. **