6 problems

Branching conjecture. Using comparisons and branching strictly enlarges the set of polynomials that can be evaluated accurately.

Let $f(x_1,\ldots,x_n)$ be a homogeneous polynomial and let $\mathcal D$ be as in the preceding conjecture. Thus condition $(A)$ means that $f$ can be written as … where each $f_m$…

Let $f(x_1,\ldots,x_n)$ be a homogeneous polynomial evaluated on a domain $\mathcal D$ with $\mathcal D\subseteq\overline{\operatorname{int}\mathcal D}$. Say that $f$ satisfies con…

Let a neural network be trained on a piecewise-linear function using a single hidden layer, ReLU activation, mean squared error loss, and floating-point computations. Inaccuracy-ac…

Optimal precision conjecture. The $m+O(\log n)$-bit bound is optimal in the sense that $m+\Omega(\log n)$ bits of precision are necessary with probability close to one.

Let $a_1,a_2,\ldots,a_n$ be floating-point numbers, let $\pi_n=a_1a_2\cdots a_n$ be their exact product, and let $\widehat{\pi}_n$ be the value computed by successive rounded multi…