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5.5 KiB

# Russian Peasant Multiplication

From Assembly to Basic to C to Javascript!

Here are my implementations of Russian Peasant Multiplication implemented in various languages:

- 6502 Assembly Language (Both ca65 and merlin32 sources)
- Applesoft BASIC
- JavaScript (Procedural version)
- JavaScript (OOP version)

A .dsk image has been provided as an convenience.

To see how much faster the Assembly version is then the BASIC version:

```
RUN RPM.BAS
BRUN RPM.BIN
```

And enter in `123456789`

* `987654321`

respectively for A and B ...

Version | Time |
---|---|

Applesoft | 33 s |

Assembly | ~1 s |

# So what the heck is it?

An alternative algorithm to implement multiplication using only:

- bit-shifts (left and right), and
- addition.

# Algorithm

- Initialize sum <- ZERO.
- IF b is ZERO then STOP.
- IF b is ODD then ADD a to sum.
- MULTIPLY a by 2. That is, shift a
**left**once. - DIVIDE b by 2. That is, shift b
**right**once. - GOTO step 2

Paste the following program into an online C compiler

```
#include <stdio.h>
int RPM( int a, int b )
{
int sum = 0;
while( b )
{
if( b & 1 )
sum += a;
a <<= 1;
b >>= 1;
}
return sum;
}
int main()
{
return printf( "%d\n", RPM( 86, 57 ) );
}
```

# Examples

Example of "traditional" multiplication:

In base 10:

```
86
x 57
----
602
430
====
4902
```

In base 2:

```
01010110 (86)
x 00111001 (57) -+
-------- V
01010110 (86 * 1*2^0 = 86)
00000000 (86 * 0*2^1 = 172) <- wasted work, partial sum = 0
00000000 (86 * 0*2^2 = 344) <- wasted work, partial sum = 0
01010110 (86 * 1*2^3 = 688)
01010110 (86 * 1*2^4 = 1376)
01010110 (86 * 1*2^5 = 2752)
==============
01001100100110 (4902 = 86*2^0 + 86*2^3 + 86*2^4 + 86*2^5)
```

Example of Russian Peasant multiplication:

In Base 10:

```
A B B Odd? Sum = 0
86 57 Yes + A = 86
x 2 = 172 / 2 = 28 No = 86
x 2 = 344 / 2 = 14 No = 86
x 2 = 688 / 2 = 7 Yes + A = 774
x 2 = 1376 / 2 = 3 Yes + A = 2150
x 2 = 2752 / 2 = 1 Yes + A = 4902
```

In Base 2:

```
A B B Odd? Sum = 0
01010110 00111001 Yes + A = 00000001010110
010101100 00011100 No = 00000001010110
0101011000 00001110 No = 00000001010110
01010110000 00000111 Yes + A = 00001100000110
010101100000 00000011 Yes + A = 00100001100110
0101011000000 00000001 Yes + A = 01001100100110
```

In Base 8:

```
A B B Odd? Sum = 0
126 71 Yes + A = 126
x 2 = 254 / 2 = 34 No = 126
x 2 = 530 / 2 = 16 No = 126
x 2 = 1260 / 2 = 7 Yes + A = 1406
x 2 = 2540 / 2 = 3 Yes + A = 4146
x 2 = 5300 / 2 = 1 Yes + A = 11446
```

In Base 16:

```
A B B Odd? Sum = 0
56 39 Yes + A = 56
x 2 = AC / 2 = 1C No = 56
x 2 = 158 / 2 = E No = 56
x 2 = 2B0 / 2 = 7 Yes + A = 306
x 2 = 560 / 2 = 3 Yes + A = 866
x 2 = AC0 / 2 = 1 Yes + A = 1326
```

# Bases

Does this algorithm work in other bases such as 2, 8, or 16?

Consider the question:

Q. Does multipling by 2 work in other bases?

A. Yes.

Q. Why?

A. When we write a number in a different base we have the *same* **representation** but a *different* **presentation.**

Adding, subtracting, multiplying, dividing all *function* the same *regardless* of which base we use.

This is the *exact* same reason that 0.999999... = 1.0. The exact same *represented* number is *presented* differently -- which
confuses the uninformed. It is a "Mathematical illusion."

Proof:

```
1 = 1 Tautology
1/3 = 1/3 Divide by sides by 3
3 * 1/3 = 3 * 1/3 Multiply by sides by 3
3 * 1/3 = 3 * 0.333333... Express integer fraction in decimal
1 = 3 * 0.333333... Simply left side (fractions cancel)
1 = 0.999999... Simply right side
```

QED.

# Efficiency

For a "BigInt" or "BigNumber" library this *is NOT* the most efficient (*) way to
multiply numbers as it doesn't scale (**). However, it is rather trivial to implement. You only need a few
functions:

`isEven()`

`isZero()`

`Shl()`

`Shr()`

`AddTo()`

Notes:

(*) Almost everyone uses FFT (Fast Fourier Transforms), Toom, and/or Karatsuba for multiplication. For example GMP, GNU Multiple Precision arithmetic library, uses **seven** different multiplication algorithms!

- Basecase
- Karatsuba
- Toom-3
- Toom-4
- Toom-6.5
- Toom-8.5
- FFT

(**) What do we mean by "Doesn't scale"? As the input numbers uses more bits we end up doing more work other other algorithms.

# References

- https://tspiteri.gitlab.io/gmp-mpfr-sys/gmp/Algorithms.html#Multiplication-Algorithms
- https://en.wikipedia.org/wiki/Multiplication_algorithm
- Multiplication is associative
- Multiplication is commutative
- https://en.wikipedia.org/wiki/Order_of_operations