Integer Representation and Operation

Size and range of types

`char - short - int- long

Overflow

• Unsigned : x + y < x / x - y > x
• Signed: p + p = n / n + n = p
• When comparing signed with unsigned, signed is changed to unsigned and compared.

Shift

• logical: shift in 0's
• arithmetic: replicate MSB
• Diving numbers in the 2's complement system causes rounding to the next smallest integer, not towards 0 as desired.

Biasing in division by shifting

• Add 2^k-1 if x < 0
• Then shift

Binary Multiplication

• Multiplying two (n)-bit numbers yields at most a (2n) bit product.
• When signed → must sign extend partial products (out to 2n bits).

Binary Division

• Dividing two (n)-bit numbers may yield an (n)-bit quotient and (n)-bit remainder.

Floating Point

Floating Point, Base 10

1.2345 * 10^{exp}

• Bias = 4
• Stored as 12345[exp] (ex.123459 = 1.2345*10^5)
• Not associative

Fixed Point, Base 2

• Radix point assumed to be in a fixed location for all numbers.
• Floating points allows the radix point to be in a different location for each value.

Floating Point, Base 2

+- b.bbb * 2^{+- exp}

[sign] b.[frac] * 2^[exp]

• Normalized FP format: +- 1.bbbbbb * 2^{+- exp}
• Floating-point numbers are always normalized.
• The 1. is not stored but assumed since we always will store normalized numbers.

IEEE 754 Floating Point Formats

Excess-N Exponent Representation

• Instead of 2's complement
• So that comparisons x < y are simple.

Single Precision (32-bit)

• float in C
• 1 sign bit
• 8 exponent bits
  • range of exponent = -126 to +127
  • value = stored - 127
• 23 fraction bits
• Equivalent decimal range: 7 digits * 10^38
• s(1) | exp(8) | fraction(23)

Double Precision (64-bit)

• double in C
• 1 sign bit
• 11 exponent bits
  • value = stored - 1023
• 52 fraction bits
• Equivalent decimal range: 16 digits * 10^-308
• s(1) | exp(11) | fraction(52)

Special Values

• float doesn't wrap around like int

Denormalized

• 0 00000001 0000..0 is (1.0) * 2^-126 == 2^-126 (norm)
• 0 00000000 1000..0 is (0.1) * 2^-126 == 2^-127 (denorm)
• 0 00000000 0100..0 is (0.01) * 2^-126 == 2^-128 (denorm)
• Q. What exponent value is used by denormalized 32-bit floating-point numbers?
  • A: -126

12-bit "IEEE Short" Format

• 1 sign bit, 5 exponent bits (excess 15), 6 fraction bits

Rounding

Round to Nearest, Half to Even

• 10...0 : round to even
• 1x...x : round up
• 0x...x : round down

Round towards 0 (chopping)

Rounding Implementation -check

• Guard bits: bits immediately after LSB of fraction
• Round bit: bit to the right of the guard bits
• Sticky bit: Logical OR of all other bits after Guard & R bits.

FP Addition/Subtraction

• Not associative!!!
• Add similar, small magnitude numbers first

FP Multiplication/Division

• Not associative - order matters!!!
• Doesn't distribute over addition