Floating Point Arithmetic, Overflow, Underflow, Normalization, Rounding

Let $X = M_x \times 2^{E_x}$, $Y = M_y \times 2^{E_y}$ (IEEE 754: $M=1.f$ normalized).

1. Addition / Subtraction

1. Align exponents: $\Delta E = |E_x-E_y|$, shift smaller mantissa right by $\Delta E$: $M_{small}' = M_{small} /2^{\Delta E}$, $E = \max(E_x,E_y)$.
2. Add/Subtract mantissas: $M = M_x' \pm M_y'$.
3. Normalize (see below).
4. Round and re-normalize if carry.

E.g. $1.101\times2^3 + 1.011\times2^1$: Align $1.011\times2^1 =0.01011\times2^3$, sum $=1.11111\times2^3$.

Subtraction similar, may need large normalization if cancellation.

2. Multiplication

$M = M_x \times M_y$ (unsigned), $E = E_x + E_y - bias$ (127/1023), sign $S = S_x \oplus S_y$.
Product of 1.xxx numbers in $[1,4)$ so if $M \ge 2$, shift right 1, $E++$.

E.g. $1.5\times2^2 \times 1.25\times2^1$: $1.5\times1.25=1.875$, $E=3$ => $1.875\times2^3=15$.

3. Division

$M = M_x / M_y$, $E = E_x - E_y + bias$, $S = S_x \oplus S_y$.
If $M<1$, shift left 1, $E--$.

4. Overflow

Exponent after operation $> E_{max}$ (255 bias => actual 127 single /1023 double).
Result magnitude too large to represent => ±Infinity (IEEE) or exception.
E.g. Max single $\approx 3.4\times10^{38}$; multiplying two large numbers overflows.

Addition/multiplication overflow: exponent carry out; division never overflows unless divisor tiny.

5. Underflow

Exponent $< E_{min}$ ($0$ => actual -126 single). Result too close to zero (denormal) => magnitude $<1.0\times2^{-126}$.
Gradual underflow: return subnormal $0.M \times2^{-126}$ with loss of precision, or flush to $0$.
E.g. $1.0\times2^{-126} / 2 =0.5\times2^{-126}$ underflows to subnormal.

6. Normalization

Keep mantissa in $[1.0,2.0)$: leading bit 1 before binary point (hidden 1).

• Post-arithmetic normalization:

• If $M \ge 2.0$ (e.g. $10.xxx$), shift right 1, $E++$.
• If $M <1.0$ (e.g. $0.001xx$), shift left until $1.xxx$, $E--$ each shift.

Maximizes precision and ensures unique representation. Unnormalized example $0.1001\times2^3$ -> $1.001\times2^2$.

7. Rounding

After normalization mantissa may have >23 bits (guard, round, sticky bits needed). IEEE requires:

• Truncation/Chop: drop extra bits (biased).
• Round to nearest even (default IEEE): if $>0.5$ ULP round up, if $=0.5$ round to make LSB 0.
• Round toward $+\infty$ / $-\infty$ / toward 0.

Guard ($G$), Round ($R$), Sticky ($S$) bits capture discarded bits to decide rounding without full precision. Incorrect rounding causes accumulated error.

Exception flags: Overflow, Underflow, Inexact (rounding needed), Invalid (0/0), Divide-by-zero.