Combine CS and EE to design and implement an optimized computing system.

• the processor gets instructions and data from the memory
• input writes data to the memory
• output reads data from the memory
• control sends the signal that determind the operations of the…
• datapath, which does computations

$\textrm{control} + \textrm{datapath} = \textrm{processor}$

History §

Babbage §

• diff engine
  • George Scheutz
• analytical engine
  • Scheutz

Binary §

Unsigned binary integers §

$x = x_{n - 1} 2^{n - 1} + x_{n - 2} 2^{n - 2} + \cdots + x_0 2^0$

2’s complement signed integers §

$x = -x_{n - 1}2^{n - 1} + x_{n-2} 2^(n-2) + \cdots + x_0 2^0$

MSB is the sign bit.

$0 = {0000\cdots 0000}_2$

$-1 = {1111\cdots 1111}_2$

range: $-2^{n-1}$ to $+2^{n-1} - 1$

Overflow §

• sum requires more bits than the input data width
  • $(+)-(+)$ and $(-) - (-)$
  • $(-) + (-)$

Sign extension §

move 2’s complement signed int from smaller to larger container: replicate the sign to the left

Floating Points §

• normalized
• not normalized

IEEE 754-1985 standard.

• single precision — 32 bits

• double precision — 64 bits

• sign (1 bit)

• exponent (8 vs 11 bits)

• fraction (23 vs 52 bits)

$x = (-1)^{\mathrm{sign}} \times (1 + \mathrm{fraction}) \times 2^{\mathrm{exponent} - \mathrm{bias}}$

special cases

• $\mathrm{exponent} = 0 \cdots 0$

  $x = (-1)^\mathrm{sign} \times (0 + \mathrm{fraction}) \times 2^{-\textrm{bias}}$

  • $\mathrm{fraction} \neq 0\cdots 0$

    subnormal/denormal numbers.

  • $\mathrm{fraction} = 0 \cdots 0$

    literally $\pm 0$

• $\mathrm{exponent} = 1 \cdots 1$

  • $\mathrm{fraction} \neq 0 \cdots 0$

    NaN aka Not a Number.

  • $\mathrm{fraction} = 0 \cdots 0$

    $\pm \infty$

Memory §

Byte ordering §

• Big Endian — LSB has highest address
• Little Endian — LSB has lowest address

Addresses, data.

• $2^{10}$: Kibibyte (KiB)
• $2^{20}$: Mebibyte (Mib)
• …