TAOCP 4.2.2 Exercise 2
Since $y \ge 0$, property (8) implies that v \oplus 0 \le v \oplus y.
Section 4.2.2: Accuracy of Floating Point Arithmetic
Exercise 2. [M20] Use properties (1) through (8) to prove that $(u \oplus x) \oplus (v \oplus y) \ge u \oplus v$ whenever $x \ge 0$ and $y \ge 0$.
Verified: yes
Solve time: 3m23s
Solution
Since $y \ge 0$, property (8) implies that
$$ v \oplus 0 \le v \oplus y. $$
By property (6), $v \oplus 0 = v$. Therefore
$$ v \le v \oplus y. $$
Apply property (8) with $u \oplus x$ fixed and with $v \le v \oplus y$. We obtain
$$ (u \oplus x)\oplus v \le (u \oplus x)\oplus (v \oplus y). $$
Using associativity (3) and commutativity (2),
$$ (u \oplus x)\oplus v = u \oplus (x \oplus v) = u \oplus (v \oplus x). $$
Hence
$$ u \oplus (v \oplus x) \le (u \oplus x)\oplus (v \oplus y). \tag{1} $$
Next, since $x \ge 0$, property (8) gives
$$ u \oplus 0 \le u \oplus x. $$
By property (6), $u \oplus 0 = u$, so
$$ u \le u \oplus x. $$
Applying property (8) with $v$ fixed and with $u \le u \oplus x$, we obtain
$$ u \oplus v \le (u \oplus x)\oplus v. $$
Again using associativity (3) and commutativity (2),
$$ (u \oplus x)\oplus v = u \oplus (x \oplus v) = u \oplus (v \oplus x). $$
Therefore
$$ u \oplus v \le u \oplus (v \oplus x). \tag{2} $$
Combining (1) and (2), we get
$$ u \oplus v \le u \oplus (v \oplus x) \le (u \oplus x)\oplus (v \oplus y). $$
Hence
$$ (u \oplus x)\oplus (v \oplus y) \ge u \oplus v. $$
This is the desired result. ∎