floating-point.md

modified	title
2025-01-14 21:57:38 UTC	Floating Point

Floating Point

• Floating points provide accuracy of a greater range of values with a given number of bits

Components of Floating Points

• Sign bit
• Mantissa
• Exponent

General Information

• If the Exponent is positive then the decimal place moves to the left and if the exponent is negative then the decimal place moves to the right
• Always involves a 'Sign Bit'
• Always uses Two's Compliment

Converting to Floating Point

1. Determine the Sign Bit:

   • If the number is positive, the sign bit is 0.
   • If the number is negative, the sign bit is 1.

2. Convert the Number to Binary:

   • Convert the absolute value of the number to its binary form.

3. Normalize the Binary Number:

   • Adjust the binary number so that it is in the form of 1.xxxxx * 2^n.
   • Count the number of positions the decimal point has moved to determine the exponent n.

4. Calculate the Exponent:

   • Add the bias to the exponent n. The bias is typically 127 for single-precision floating point numbers.
   • Convert the biased exponent to binary.

5. Determine the Mantissa:

   • The mantissa is the fractional part of the normalized binary number (excluding the leading 1).

6. Combine the Components:

   • Combine the sign bit, the exponent, and the mantissa to form the final floating point representation.

Example:

• Convert -5.75 to floating point:
  1. Sign bit: 1 (since the number is negative)
  2. Binary: 5.75 in binary is 101.11
  3. Normalize: 1.0111 * 2^2
  4. Exponent: 2 + 127 = 129 → 10000001 in binary
  5. Mantissa: 01110000000000000000000
  6. Final representation: 1 10000001 01110000000000000000000