TAOCP 4.2.2 Exercise 6
By equation (6) of the floating point arithmetic defined in this section, 0 \oplus u = u for every floating point number $u$.
Section 4.2.2: Accuracy of Floating Point Arithmetic
Exercise 6. [M22] Are either of the following two identities valid for all floating point numbers $u$? (a) $0 \oplus (0 \oplus u) = u$; (b) $1 \otimes (1 \otimes u) = u$.
Verified: yes
Solve time: 3m40s
Solution
(a) $0 \oplus (0 \oplus u) = u$
By equation (6) of the floating point arithmetic defined in this section,
$$ 0 \oplus u = u $$
for every floating point number $u$.
Therefore
$$ 0 \oplus (0 \oplus u) = 0 \oplus u = u. $$
Hence the identity
$$ 0 \oplus (0 \oplus u)=u $$
is valid for all floating point numbers $u$.
(b) $1 \otimes (1 \otimes u) = u$
By the definition of floating point multiplication,
$$ 1 \otimes u = \operatorname{fl}(1\cdot u), $$
where $\operatorname{fl}(x)$ denotes the floating point number obtained by rounding the exact value $x$.
If $u$ is a floating point number, then the exact product satisfies
$$ 1\cdot u = u. $$
Since $u$ is already a floating point number, rounding leaves it unchanged:
$$ \operatorname{fl}(u)=u. $$
Consequently,
$$ 1 \otimes u = u $$
for every floating point number $u$.
Applying the same identity once more,
$$ 1 \otimes (1 \otimes u) = 1 \otimes u = u. $$
Therefore
$$ 1 \otimes (1 \otimes u)=u $$
for all floating point numbers $u$.
Both identities are valid:
$$ \boxed{0 \oplus (0 \oplus u)=u} $$
and
$$ \boxed{1 \otimes (1 \otimes u)=u}. $$