TAOCP 7.1.3 Exercise 97
The proposed solution does not address the problem stated in Exercise 7.
Section 7.1.3: Bitwise Tricks and Techniques
Exercise 97. [23] Leslie Lamport's paper in 1975 presented the following "problem taken from an actual compiler optimization algorithm": Given octabytes $x = (x_7 \ldots x_0){256}$ and $y = (y_7 \ldots y_0){256}$, compute $t = (t_7 \ldots t_0){256}$ and $z = (z_7 \ldots z_0){256}$ so that $t_j \ne 0$ if and only if $x_j \ne 0$, $x_j \ne \texttt{''}$, and $x_j \ne y_j$; and $z_j = (x_j = 0?$ $y_j$: $(x_j \ne \texttt{''} \land x_j \ne y_j ?$ $\texttt{'*'}$ : $x_j)$).
Verified: no
Solve time: 12m33s
Correctness
The proposed solution does not address the problem stated in Exercise 7.1.3.97. The exercise requires construction of two bytewise functions $t$ and $z$ involving three cases: $x_j = 0$, $x_j = \texttt{'*'}$, and $x_j = y_j$. The solution instead derives a subtraction identity and a bytewise comparison test unrelated to the required conditional structure. None of the required expressions for $t_j$ or $z_j$ are constructed or verified.
Gaps and Errors
The entire argument is off-topic relative to the specification of the exercise, which is a critical error. The problem requires character-based conditional logic on octabytes with a distinguished symbol $\texttt{'*'}$, but the solution never introduces or uses this symbol.
The derivation of
$x - y = (x \oplus y) - 2(\bar{x} ,&, y)$
is unrelated to the exercise and does not contribute to computing either $t$ or $z$.
The construction of
$t_j = (y_j - x_j - 1) ,&, 128$
depends on interpreting $x_j \le y_j$, which is not part of the required condition. The required predicate is $x_j \ne 0$, $x_j \ne \texttt{'*'}$, and $x_j \ne y_j$, which cannot be reduced to a single inequality test between $x_j$ and $y_j$.
The proposed $z_j$ formula is entirely absent, so the main output specification is not satisfied.
Summary
The solution fails to engage with the actual problem statement and instead solves a different bytewise comparison problem.
VERDICT: FAIL, the solution does not address the required conditional construction for $t_j$ and $z_j$.