Scaling numbers cleverly to prevent arithmetic overflows or rounding errors

Let's say I have an integer position $p$ inside some integer range $r$, such that

$$0 \le p < r \;\;\;\textrm{with}\;p, r \in \mathbb{N}$$

I need to scale a position $p$ from a small range where $0<r<2^{31}$ to the corresponding position $P$ in a big range $0<R<2^{63}$ and back (in programming terms: I have one scenario where my position and range are 32-bit integers and another where my position and range are 64-bit integers).

So, essentially all I need to do is:

$$P = \lfloor \, p * R / r\rfloor\;\;\;\;\;\textrm{and}\;\;\;\;\ p = \lfloor P * r / R \rfloor$$ But, for this transformation, I am only allowed to perform 64-bit integer operations.

To scale the small range up to the big range without generating overflows or rounding errors is relatively easy:

$$P = \lfloor \, p * R / r\rfloor = \lfloor R / r\rfloor * p + \lfloor((R\bmod{r} ) * p) / r\rfloor$$ (I used $\lfloor\cdot\rfloor$ to indicate the implicit rounding of the integer division operations)

Aside - a quick explanation of the reasoning behind this equation: Since $p<r$, the first term in the sum will never exceed $2^{63}$. But, doing the division "first" to make this happen, leads to a rounding error. This error is captured by the modulus operation in the second term, which, since $p<r<2^{31}$, can also not exceed $2^{63}$. Thus, this formula allows me to calculate $P$ such that it is correctly rounded down to the nearest integer, only using integer arithmetic with numbers $<2^{63}$.

What I'm having some difficulty with is finding the reverse formula, i.e. for $p = \lfloor P * r / R \rfloor$:

I'm looking for a function $f(P, r, R)$ that calculates $\lfloor P * r / R \rfloor$ while only using 64-bit integer arithmetic, i.e. every division implies an immediate rounding down and all intermediate values must fall within the range $[{-\,2^{63}, +\,2^{63}}[$.

UPDATE:

I've been playing with a recursive solution by partially solving the reverse equation for $p$:

$$P = \lfloor R/r \rfloor * p + \lfloor ((R\bmod{r})*p)/r \rfloor$$

$$\lfloor R/r \rfloor * p_{n+1} = P - \lfloor ((R\bmod{r})*p_n)/r \rfloor$$

$$p_{n+1} = \bigg\lfloor \frac{P}{\lfloor R/r \rfloor} \bigg\rfloor - \bigg\lfloor \frac{\lfloor ((R\bmod{r})*p_n)/r \rfloor} {\lfloor R/r \rfloor} \bigg\rfloor$$

For most values of $r$, $P$, and $R$ it very quickly gives a value that is within $\pm 1$ of the actual answer. But sometimes, it ends up oscillating between two consecutive values and if $R\bmod{r}$ becomes much larger than $R/r$, convergence is either really slow, or sometimes doesn't happen at all.

Can this be improved somehow to guarantee quick convergence to the actual answer? Or maybe it's possible to make this non-recursive?

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Can you perform $\left\lfloor a\cdot b\right\rfloor\bmod2^{63}$ even if $a\cdot b$ would exceed the 64 bit barrier? In other words, can you rely on wraparound semantics for integer multiplication overflow? – MvG Jul 2 '13 at 17:20
@MvG Yep, should be no problem. Since I need to use (long-)integers to do the math, though, they might wrap to a negative value first. I am trying to design an algorithm to implement with a computer, so all the usual (long-)integer behavior applies. (I asked a corresponding, more coding-oriented version of this question on stackoverflow as well, but haven't had any luck with responses...) – Markus A. Jul 2 '13 at 17:53
Found your question on SO. I guess I'd try to interpret all numbers in base 2<sup>32</sup>. You should be able to obtain a 3-digit dividend and a 2-digit divisor, on which you can perform long division. Is this hint enough? – MvG Jul 2 '13 at 18:09
@MvG Basically implementing a specialized version of a BigInteger library like GMP (en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library)... I had been looking into that. I just wanted to do some research and thinking to see whether I could find a clever trick instead that would yield the same result more elegantly with fewer math ops. – Markus A. Jul 2 '13 at 18:38
Directly implementing long division in base $2^{64}$ is almost surely your best bet. Base $2^{32}$ is probably your second best bet. – Hurkyl Jul 4 '13 at 8:58

Disclaimer: This isn't a 100% complete answer, but for the case that I need it for, it works. Maybe someone with additional ideas can improve it...

One can make some progress by looking at different possible special cases:

If $P*r<2^{63}$, there is no problem with overflows to begin with, so we can just calculate the answer directly as $p=\lfloor P*r/R \rfloor$. The only thing we need to do is to perform the check as $P<\lfloor2^{63}/r \rfloor$ instead to allow for a false result without overflow.

To cover the other case, one can do the following transformation:

$$p=\lfloor P*r/R \rfloor= \bigg\lfloor\frac{P}{R/r}\bigg\rfloor=\bigg\lfloor\frac{P}{\lfloor R/r \rfloor + (R\bmod{r})/r}\bigg\rfloor$$

$$=\bigg\lfloor\frac{P}{\lfloor R/r \rfloor}-\frac{P(R\bmod{r})/r}{\lfloor R/r \rfloor(R/r)}\bigg\rfloor=\Bigg\lfloor\bigg\lfloor\frac{P}{\lfloor R/r\rfloor}\bigg\rfloor + \frac{P\bmod{\lfloor R/r\rfloor}}{\lfloor R/r\rfloor}-\frac{P(R\bmod{r})}{R\lfloor R/r \rfloor}\Bigg\rfloor$$

$$=\bigg\lfloor\frac{P}{\lfloor R/r\rfloor}\bigg\rfloor + \Bigg\lfloor\frac{P\bmod{\lfloor R/r\rfloor}}{\lfloor R/r\rfloor}-\frac{P}{R}\frac{R\bmod{r}}{\lfloor R/r \rfloor}\Bigg\rfloor$$

The first term in the sum can be calculated directly without overflow issues. The second half after the '$+$' - sign is a little more difficult. But, looking at it, one realizes the following: The fraction to the left of the '$-$' - sign is always between $\ge 0$ and $<1$. Also: $0 \le P/R < 1$. So, we can look at two different special cases here:

1. If $R\ge r^2$, the entire part to the right of the '$+$' - sign can only be either $0$ or $-1$.
2. Otherwise, things could get more complicated...

Luckily, in my application, I know for a fact, that $R\ge r^2$ is always true. Therefore, for me, the answer is:

$$p = \left\{\begin{array}{1 1}\lfloor (P*r)/R \rfloor & \quad \text{if P<\lfloor2^{63}/r \rfloor} \\ \lfloor P/\lfloor r/R\rfloor \rfloor \;[-1] & \quad \text{if P\ge \lfloor2^{63}/r \rfloor and R\ge r^2} \\ \text{here be dragons} & \quad \text{if P\ge \lfloor2^{63}/r \rfloor and R<r^2} \end{array} \right.\$$

Here, $[-1]$ indicates that I only subract -1 in some cases, which I check by first leaving it out, testing the value of $p$ I get and then putting it in if I need it.

In the third case, one could use an arbitrary precision approach (c.f. GMP) and calculate the answer by doing long division as suggested by MvG. But maybe someone can find a more elegant way to slay the dragons for completeness... ;) Otherwise, if I have time a little later, I (or someone else) might add the formula for implementing the long division.

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