IEEE 754 Floating Point Standard

Represents real numbers in binary scientific notation: $(-1)^S \times (1.F) \times 2^{E-bias}$ for normalized numbers.

Single Precision (32-bit):

| S (1) | Exponent (8) | Mantissa/Fraction (23) |

• $S=0$ positive, $1$ negative.
• Exponent = true exponent $E$ + bias $127$ (range $-126$ to $+127$).
• Mantissa stores fractional part after leading 1. (hidden bit).
• Value $= (-1)^S \times 1.M \times 2^{E-127}$.

Double Precision (64-bit): $S(1)$ | Exponent(11, bias 1023) | Mantissa(52). Value $= (-1)^S \times 1.M \times 2^{E-1023}$.

Special values:

• $E=0, M=0$ → $\pm0$; $E=0, M
  eq0$→ denormalized$0.M \times 2^{-126}$ (gradual underflow).
• $E=255, M=0$ → $\pm\infty$; $E=255, M
  eq0$ → NaN.
• Normalized: $1\le E\le254$.

Example - Single Precision

Represent $-12.75_{10}$ in IEEE 754 32-bit:

1. Sign: negative → $S=1$.
2. Binary: $12.75 = 1100.11_2 = 1.10011 \times 2^{3}$ (normalize).
3. Exponent: $E_{true}=3$ → $E_{biased}=3+127=130=10000010_2$.
4. Mantissa: fractional part 10011 padded to 23 bits → 10011000000000000000000.

Result:

1 | 10000010 | 10011000000000000000000

Hex = 0xC14C0000.

Another example - $0.085$? $6.5_{10}=110.1_2=1.101\times2^{2}$ → $S=0, E=129=10000001, M=1010...$ → 0 10000001 10100000000000000000000 = 0x40D00000.

To decode, reverse: $S$, $E-127$, insert 1., compute $(-1)^S\times1.M\times2^{E-bias}$.