Subtract with carry

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Subtract-with-carry is a pseudorandom number generator: one of many algorithms designed to produce a long series of random-looking numbers based on a small amount of starting data. It is of the lagged Fibonacci type introduced by George Marsaglia and Arif Zaman in 1991.^[1] "Lagged Fibonacci" refers to the fact that each random number is a function of two of the preceding numbers at some specified, fixed offsets, or "lags".

Sequence generated by the subtract-with-carry engine may be described by the recurrence relation:

${\displaystyle x(i)=(x(i-S)-x(i-R)-cy(i-1))\ {\bmod {\ }}M}$

where ${\displaystyle cy(i)={\begin{cases}1,&{\text{if }}x(i-S)-x(i-R)-cy(i-1)<0\\0,&{\text{otherwise}}\end{cases}}}$.

Constants S and R are known as the short and long lags, respectively.^[2] Therefore, expressions ${\displaystyle x(i-S)}$ and ${\displaystyle x(i-R)}$ correspond to the S-th and R-th previous terms of the sequence. S and R satisfy the condition ${\displaystyle 0<S<R}$. Modulus M has the value ${\displaystyle M=2^{W}}$, where W is the word size, in bits, of the state sequence and ${\displaystyle W>0}$.

The subtract-with-carry engine is one of the family of generators which includes as well add-with-carry and subtract-with-borrow engines.^[1]

It is one of three random number generator engines included in the standard C++11 library.^[3]