# 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](rpm_ca65.s) and [merlin32](rpm_m32.s) 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
1. Initialize Sum <- zero. In C nomenclature: `Sum = 0;`
2. If B is odd then add A to Sum. In C nomenclature: `if( B & 1 ) Sum += A;`
3. Multiply A by 2 -- that is, Shift A **left** by one. In C nomenclature: `A <<= 1;`
4. Divide B by 2 -- that is, Shift B **right** by one. In C nomenclature: ` B >>= 1;`
5. If B is zero then STOP. `while( b ) { ... }`
6. Goto step 2
Paste the following program into an [online C compiler](https://www.onlinegdb.com/online_c_compiler)
```c
#include
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)
00111001 (57)
--------
01010110 (86 * 2^0 = 86)
00000000 (86 * 2^1 = 172) <- wasted work, partial sum = 0
00000000 (86 * 2^2 = 344) <- wasted work, partial sum = 0
01010110 (86 * 2^3 = 688)
01010110 (86 * 2^4 = 1376)
01010110 (86 * 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](https://gmplib.org/manual/), GNU Multiple Precision arithmetic library, uses **[seven](https://gmplib.org/manual/Multiplication-Algorithms.html#Multiplication-Algorithms)** 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