Floating points

Floating point representation

Numerical form

$-1^s\times M\times 2^E$ s: Sign bit / 0 - non-negative, 1 - negative M: Significand M / normally a fractional value in range [1.0, 2.0) E: Exponent E / weights value by power of two

Precisions

Single precision - 32 bits

• 8 exp bits, 23 frac bits Double precision - 64 bits
• 11 exp bits, 52 frac bits Extended precision - 80 bits
• 15 exp bits, 63 frac bits
• 1 bit wasted

Normalized Values

Condition

• exp != 000…0
• exp != 111…1 Exponent coded as a biased value
• E = Exp - Bias
• Bias : Bias value (=$2^{k-1}-1$, where k is the number of exp bits)
  • Single precision ) 127
  • Double precision ) 1023 Significand coded with with implied leading 1

Example

float f = 2003.0

Denormalized Values

Condition

• exp = 000…0 Value
• Exponent value E = 1 - Bias
• Significand value M = 0.xxx…x No implied leading 1 Case 1 - exp = 000…0, frac = 000…0
• Represents value 0.0
• Two zeros (+0, -0) Case 2 - exp = 000…0, frac != 000…0
• Numbers very close to 0.0
• For gradual underflow / Otherwise, the value su
• Gradual underflow : possible numeric values are spaced evenly near 0.0

Special Values

Condition

• exp = 111…1 Case 1 - exp = 111…1, frac = 000…0
• Infinity Case 2 - exp = 111…1, frac != 000…0
• Not-a-number