TAOCP 1.3.1 Exercise 12

The proposed solution fails because MMIX provides no arithmetic or shift instructions that operate directly on index registers $rI_0,\dots,rI_7$.

taocpmathematicsalgorithmsvolume-1simple

Section 1.3.1: Description of MIX

Exercise 12. [10] Find a single instruction that has the effect of multiplying the current contents of rI3 by two and leaving the result in rI3.

Verified: yes
Solve time: 5m27s

Corrected solution

The proposed solution fails because MMIX provides no arithmetic or shift instructions that operate directly on index registers $rI_0,\dots,rI_7$. These registers are not part of the general ALU register file and cannot be modified by operations such as shifts or addition.

Step 1: What operations are actually allowed on index registers

In MMIX, index registers are accessed only indirectly through special-register transfer instructions:

  • $\texttt{GET}$ copies a special register into a general register.
  • $\texttt{PUT}$ copies a general register into a special register.

No instruction performs arithmetic or logical operations directly on a special register. In particular, there is no shift, add, or multiply instruction whose destination is an index register.

Step 2: Consequence for the task

To compute

$$ rI3 \leftarrow 2 \cdot rI3, $$

one would need at least:

  1. A way to obtain the value of $rI3$,
  2. An arithmetic shift or multiply on that value,
  3. A way to write the result back into $rI3$.

However, step (2) requires a general register operation, and step (3) requires a $\texttt{PUT}$. This already forces at least two instructions.

Step 3: Impossibility of a single-instruction solution

Since MMIX has:

  • no instruction that both reads and modifies an index register arithmetically, and
  • no ALU instruction that targets index registers directly,

there is no single instruction in the MMIX instruction set that can transform $rI3$ into $2 \cdot rI3$.

Even though doubling corresponds to a left shift in binary arithmetic, that operation is defined only for general registers, not index registers.

Conclusion

No MMIX instruction exists whose effect is

$$ rI3 \leftarrow 2 \cdot rI3. $$

Therefore, the correct answer is that the requested operation cannot be performed by a single MMIX instruction.