IEEE 754 Floating Point Numbers

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IEEE 754 → Decimal

• An IEEE 754 floating point number consists of 3 parts, the sign, exponent and the mantissa.
• The decimal number can be constructed with the equation .
  • Where is the sign, is the exponent, and is the mantissa.
• Consider the binary number 0 10000101 01010100100000000000000

Sign Bit

• The sign bit is the first part of an IEEE 754 floating point number.
• For instance, with a binary number of the example would be 0, which would mean it’s a positive number.

Exponent

• The exponent indicates how big the floating point number will be.
• IEEE 754 uses a bias of 127, so you subtract your number from that.
  • The bias is so the exponent can also support negative numbers
• In this case, 10000101 is binary for 133.
  • The actual exponent would be

Mantissa (Significand)

• The mantissa is what stores the actual ‘digits’ of the decimal number.
• The mantissa has an implicit leading 1
• In this case, it’s 1010100100000000000000, which can be shorted to 10101001
  • With the implicit leading 1:

Final Decimal Value

• The final decimal value is the mantissa multiplied by the exponent.
• In our case,

Decimal → IEEE 754

• To convert a decimal to IEEE 754, it’s pretty straightforward—you just need the 3 parts.
• For instance, consider a decimal number 13.625

Sign Bit

• Since 13.625 is positive, the sign bit would be 0

Mantissa

• We need to split 13.625 into the whole and fractional parts: 13.625 → 1101.101
• Once we have the two parts, we need to normalize it by the decimal point to the left.
  • Each position you shift it is one power towards the exponent
• The normalized mantissa (with leading 1) would be 1.101101

Exponent

• Since we had to normalize the decimal point by moving it places, the exponent would be .
• Since the bias is , the actual exponent value would be ()

Final IEEE 754

• Once you have these three parts, you can combine them to get the final float.
  • Sign → 0
  • Exponent → 10000010
  • Mantissa → 10110100000000000000000 (no leading 1)
• 13.625 → 0 10000010 10110100000000000000000

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