fbpx

C++: Ethiopian Multiplication

Bjarne-stroustrup
 

A method of multiplying integers using only addition, doubling, and halving.

Method:

  1. Take two numbers to be multiplied and write them down at the top of two columns.
  2. In the left-hand column repeatedly halve the last number, discarding any remainders, and write the result below the last in the same column, until you write a value of 1.
  3. In the right-hand column repeatedly double the last number and write the result below. stop when you add a result in the same row as where the left hand column shows 1.
  4. Examine the table produced and discard any row where the value in the left column is even.
  5. Sum the values in the right-hand column that remain to produce the result of multiplying the original two numbers together

For example: 17 × 34

       17    34

Halving the first column:

       17    34
        8
        4
        2
        1

Doubling the second column:

       17    34
        8    68
        4   136 
        2   272
        1   544

Strike-out rows whose first cell is even:

       17    34
        8    68 
        4   136 
        2   272 
        1   544

Sum the remaining numbers in the right-hand column:

       17    34
        8    -- 
        4   --- 
        2   --- 
        1   544
           ====
            578

So 17 multiplied by 34, by the Ethiopian method is 578.

The task is to define three named functions/methods/procedures/subroutines:

  1. one to halve an integer,
  2. one to double an integer, and
  3. one to state if an integer is even.

Use these functions to create a function that does Ethiopian multiplication.

 

Using C++ templates, these kind of tasks can be implemented as meta-programs. The program runs at compile time, and the result is statically saved into regularly compiled code. Here is such an implementation without tutor, since there is no mechanism in C++ to output messages during program compilation.

template<int N>
struct Half    
{              
	enum { Result = N >> 1 };
};                               

template<int N>
struct Double  
{              
	enum { Result = N << 1 };
};                               

template<int N>
struct IsEven  
{              
	static const bool Result = (N & 1) == 0;
};

template<int Multiplier, int Multiplicand>
struct EthiopianMultiplication
{
	template<bool Cond, int Plier, int RunningTotal>
	struct AddIfNot
	{
		enum { Result = Plier + RunningTotal };
	};
	template<int Plier, int RunningTotal>
	struct AddIfNot <true, Plier, RunningTotal>
	{
		enum { Result = RunningTotal };
	};

	template<int Plier, int Plicand, int RunningTotal>
	struct Loop
	{
		enum { Result = Loop<Half<Plier>::Result, Double<Plicand>::Result,
			AddIfNot<IsEven<Plier>::Result, Plicand, RunningTotal >::Result >::Result };
	};
	template<int Plicand, int RunningTotal>
	struct Loop <0, Plicand, RunningTotal>
	{
		enum { Result = RunningTotal };
	};

	enum { Result = Loop<Multiplier, Multiplicand, 0>::Result };
};

#include <iostream>

int main(int, char **)
{
	std::cout << EthiopianMultiplication<17, 54>::Result << std::endl;
	return 0;
}

SOURCE

Content is available under GNU Free Documentation License 1.2.