Computer Arithmetic

Carry = 1 if ’1’ bit gets carried out of MSB. Indicates error in unsigned no. system.
V = 1 if

• A+B, MSB(A) = MSB(B) $\neq$ MSB(Result)

• A-B, MSB(A) $\neq$ MSB(B) and MSB(Result) $\neq$ MSB(A)

Sign extension: copy MSB to all the higher bits.
Multi-precision arithmetic: add the lower 32-bit words, then ADC the upper 32-bit words.
Range of 2’s comp = $-2^{N-1}$ to $2^{N-1}-1$
Fixed point

• Precision = value of LSB (smallest fractional value)

• $2^3\,2^2\,2^1\,2^0\,.\,2^{-1}\,2^{-2}\,2^{-3}\,2^{-4}$, precision is $2^{-4}$.

• Precision is fixed and uniform throughout the range.

• For a fixed number of bits, allocating more bits for the integer part (increase range) will reduce the precision.

Floating point

• 32-bit: 1 (Sign), 8 (exponent, $E$), 23 (fraction, $F$), bias = 127

• 64-bit: 1 (Sign), 11 (exponent, $E$), 52 (fraction, $F$), bias = 1023

• Format: $(-1)^S \times (1.F)_2 \times 2^{E - \text{Bias}}$

• E.g. 1 00001100 010100....0 $\equiv$ $-1.3125\times 2^{-115}$

• Normalised mode

  • Maintain a non-zero digit (1) before the radix point.

  • Exponent range: 0000 0001 to 1111 1110

  • Smallest magnitude: X 0000 0001 0000....0 $\equiv$ $1 \times 2^{-126}$

  • Largest magnitude: X 1111 1110 1111 ....1 $\equiv$ $\approx 2 \times 2^{127} = 2^{128}$

  • Underflow: numbers between $-2^{126}$ to $2^{126}$, i.e. values close to zero that cannot be represented.

Fixed vs floating pt (32-bit)

• Max range (fixed) = $2^{32}$ (no fractional part)

• Max range (floating) = $2*2^{128}$ ($-2^{128}$ to $\sim 2^{128}$)

• Max precision (fixed) = $2^{-32}$ (no integer part)

• Max precision (floating, near to 0) = less than $2^{-126}$

• Floating point has higher range and better precision at small numbers, but very coarse precision at the 2 ends of the range. Floating point provides uniform precision across entire range.

More stuff

• Arithmetic between numbers with huge diff in magnitude: add/subtract numbers of similar smaller magnitude first, to allow their magnitude to be closer to the bigger magnitudes.

• To preserve precision AMAP, do division last, as division truncates LSB and loses bits.