Dimitre Novatchev wrote on May 4, 2002:

In his excellent book Simon Thompson defines scaling of the elements of a sequence of random numbers from the interval [0, 65535] to an interval [s,t] in the following way (top of page 368):

...

scaleSequence :: Int -> Int -> [Int] -> [Int] scaleSequence s t = map scale where scale n = n `div` denom + s range = t - s + 1 denom = modulus `div` range where modulus = 65536 However, the following expression:

e4000 = (scaleSequence 1 966 ([65231] ++ [1..965]))!!0 evaluates to 974 -- clearly outside of the specified interval.

Here's the algorithm for mapping of ranges of integers that is provably correct. We aim to find a function sc(n) defined over integers 0 <= n <= M so that sc(0) -> s (given integral number) sc(M) -> t (given integral number), t>=s and 0<=a < b ==> sc(a) <= sc(b) Such a function sc(n) is given by the following expression: sc(n) = (n*(t-s)) `div` M + s Proof: sc(0) = s sc(M) = (M*(t-s)) `div` M + s = t a<b ==> a*(t-s) < b*(t-s) ==> (a*(t-s)) `div` M <= (b*(t-s)) `div` M The assertion that 0<=a<b ==> a `div` M <= b `div` M for M >0 can easily be proven by contradiction. Indeed, given the facts (a `div` M)* M = a - ra, 0<= ra <= M-1 (b `div` M)* M = b - rb, 0<= rb <= M-1 and an assumption a `div` M > b `div` M we see a = M*(a `div` M) + ra = M*(b `div` M + x) + ra { where x >= 1 } = b - rb + ra + M*x >= min{ b - rb + ra + M*x over ra in [0,M-1], rb in [0,M-1]} >= b - (M-1) + M*x >= b + 1 + M*(x-1) { x >= 1 } > b {contradiction} Thus the correct algorithm reads

scaleSequence :: Int -> Int -> [Int] -> [Int] scaleSequence s t = map scale where scale n = (n*range) `div` maxn + s range = t - s maxn = modulus - 1 modulus = 65536

Given your example, Main> any (==966) (scaleSequence 1 966 ([65231] ++ [1..965] ++[65565])) True Main> any (>966) (scaleSequence 1 966 ([65231] ++ [1..965] ++[65565])) False