Division by Integers Using Multiplication

1
Division by Integers Using Multiplication

• Chuck Pheatt
• Emporia State University

2
A More Precise Title

• Division by Invariant Integers using
  Multiplication.
• Reference Granlund, T. and Montgomery, P.L.
  Proceedings of the ACM SIGPLAN '94 Conference on
  Programming Language Design and Implementation
  (PLDI), pages 61--72, Orlando, Florida, June 1994.

3
Unsigned Integer Division

• Suppose we want to perform an unsigned integer
  division
• Where
• 0 lt d lt 2N, d is the divisor and is invariant
• 0 n lt 2N, n is the dividend and is variable
• N is the word size

4
Unsigned Integer Division

• We would like to calculate 1/d using only
  multiplications and shifts. We will use a
  multiplier of the form
• That is

5
The Unsigned Algorithm

• From Granlund and Montgomery, given a divisor d
  and dividend n

6
So.

• Processors generally have very fast integer
  multiply operations (59 cycles signed, 48
  cycles unsigned).
• The integer division instruction delivers only
  one bit of quotient per cycle (2247 cycles
  signed, 1741 cycles unsigned).

7
So.

• The cost for invariant integer division
  (per quotient)
• 1 multiply (4 cycles)
• 2 adds/subtracts (2 4 8 cycles)
• 2 shifts (2 4 8 cycles)
• 20 cycles total

8
The Activities

• A CS0 activity that introduces the idea of ones
  and twos complement numeric representation and
  using shift operations to replace multiplication
  and division operations.
• A CS1 activity that has students perform a
  computer performance assessment utilizing the
  generation of random values.
• A CS2 activity that has students use inline
  assembly language to improve computer algorithm
  performance.
• A systems programming (assembly language)
  activity where students will use the approach to
  optimize code.
• A compiler writing course activity where students
  develop a C/C program to create the code
  necessary to implement the algorithm.

9
The Activities

• The Whole Thing
• Activity 0
• Activity 1
• Activity 2
• Activity 3
• Activity 4

10

• Questions?