TAOCP 1.3.1 Exercise 7

Let $D$ be the signed integer formed from $(rA,rX)$, and let $m\neq 0$ be the signed integer in $M$.

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Section 1.3.1: Description of MIX

Exercise 7. [M15] Give a precise definition of the results of the DIV instruction for all cases in which overflow does not occur, using the algebraic operations X mod Y and \lfloor X/Y \rfloor.

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Let $D$ be the signed integer formed from $(rA,rX)$, and let $m\neq 0$ be the signed integer in $M$. Assume no overflow occurs, so that the quotient and remainder are representable MIX words.

The goal is to define the results of DIV using $\lfloor X/Y \rfloor$ and $X \bmod Y$, with a precise meaning of “mod” consistent with MIX (remainder has the sign of $D$).

1. Start from floor division and correct it to MIX quotient

Define the preliminary quotient

$$ q_0 = \left\lfloor \frac{D}{m} \right\rfloor. $$

This is the standard real-floor quotient, but it does not always match MIX semantics when $D$ and $m$ have opposite signs, because MIX requires truncation toward zero.

Now define the corresponding preliminary remainder

$$ r_0 = D - m q_0. $$

This satisfies $D = m q_0 + r_0$, but its sign is not guaranteed to match the sign of $D$.

2. Correct the quotient to enforce MIX remainder sign rule

MIX requires the remainder to satisfy:

  • $D = mq + r$
  • $|r| < |m|$
  • $\operatorname{sgn}(r) = \operatorname{sgn}(D)$ when $D \neq 0$

If $r_0 = 0$, no correction is needed:

$$ q = q_0,\quad r = 0. $$

If $r_0 \neq 0$, check whether $r_0$ already has the correct sign:

  • If $\operatorname{sgn}(r_0) = \operatorname{sgn}(D)$, set

$$ q = q_0,\quad r = r_0. $$

  • Otherwise, adjust the quotient by one step in the direction of $m$:

$$ q = \begin{cases} q_0 - 1 & \text{if } m > 0,\ q_0 + 1 & \text{if } m < 0, \end{cases} \qquad r = D - mq. $$

This correction ensures that $r$ has the same sign as $D$, while preserving $D = mq + r$.

3. Definition of MIX modulo

We now define a signed modulus consistent with MIX behavior:

$$ D \bmod m := D - m q, $$

where $q$ is the corrected quotient defined above.

Thus, by construction,

$$ r = D \bmod m. $$

This is not the Euclidean modulus. It is the MIX modulus, characterized by:

  • $D = m(D \div m) + (D \bmod m)$
  • $|D \bmod m| < |m|$
  • $\operatorname{sgn}(D \bmod m) = \operatorname{sgn}(D)$ if $D \neq 0$

4. Final definition of DIV in required notation

We can now state the result of DIV purely using $\lfloor X/Y \rfloor$ and $\bmod$.

Quotient

Let

$$ q_0 = \left\lfloor \frac{D}{m} \right\rfloor. $$

Define

$$ q = \begin{cases} q_0 & \text{if } r_0 = D - m q_0 = 0 \text{ or } \operatorname{sgn}(r_0) = \operatorname{sgn}(D),\[6pt] q_0 - \operatorname{sgn}(m) & \text{otherwise.} \end{cases} $$

Equivalently, this is truncation toward zero.

Remainder

$$ r = D \bmod m = D - mq. $$

5. Register assignment

The instruction DIV m produces:

$$ rA \leftarrow q, \quad rX \leftarrow r, $$

where $q$ and $r$ are defined above.

6. Conclusion

The MIX DIV instruction (in all non-overflow cases) is precisely defined by:

  • quotient $q$, obtained from $\left\lfloor D/m \right\rfloor$ with a single correction step to enforce truncation toward zero,
  • remainder $r = D \bmod m = D - mq$,

satisfying:

$$ D = mq + r,\quad |r| < |m|,\quad \operatorname{sgn}(r) = \operatorname{sgn}(D). $$

This provides a complete definition in terms of $\lfloor X/Y \rfloor$ and $X \bmod Y$, as required.