Question $1\mathrm{.}18.($Hard$)$ Suppose that a and $b$ are normalized IEEE double precision floating point numbers, and consider the following algorithm, running with IEEE arithmetic:

$)<mi>(</mi><mi>|</mi><mi>b</mi><mi>|</mi>$

Prove the following facts:

Barring overflow or underflow, the only roundoff error committed in running the algorithm is computing $s_1=fl(a+b).$ In other words, both subtractions $s_1-$a and $(s_1-a)-b$ are computed exactly.

$s_1+s_2=a+b,$ exactly. This means that $s_2$ is actually the roundoff error committed when rounding the exact value of $a+b$ to get $s_1.$