## Section: New Results

### Floating-point and Validated Numerics

#### Optimal bounds on relative errors of floating-point operations

Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function $fl$ and barring underflow and overflow, such models bound the relative errors ${E}_{1}\left(t\right)=|t-fl\left(t\right)|/\left|t\right|$ and ${E}_{2}\left(t\right)=|t-fl\left(t\right)|/\left|fl\left(t\right)\right|$ by the unit roundoff $u$. In [10] we investigate the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real $t$ and in the case where $t$ is the exact result of an arithmetic operation on some floating-point numbers. We show that ${E}_{1}\left(t\right)$ and ${E}_{2}\left(t\right)$ are optimally bounded by $u/(1+u)$ and $u$, respectively, when $t$ is real or, under mild assumptions on the base and the precision, when $t=x\pm y$ or $t=xy$ with $x,y$ two floating-point numbers. We prove that while this remains true for division in base $\beta >2$, smaller, attainable bounds can be derived for both division in base $\beta =2$ and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.

#### On various ways to split a floating-point number

In [32] we review several ways to split a floating-point number,
that is, to decompose it into the exact sum of two floating-point numbers of smaller precision.
All the methods considered here involve only a few IEEE floating-point operations,
with rounding to nearest and including possibly the fused multiply-add (FMA).
Applications range from the implementation of integer functions
such as `round` and `floor`
to
the computation of suitable scaling factors aimed, for example,
at avoiding spurious underflows and overflows when implementing functions such as the hypotenuse.

#### Algorithms for triple-word arithmetic

Triple-word arithmetic consists in representing high-precision numbers as the unevaluated sum of three floating-point numbers. In [45], we introduce and analyze various algorithms for manipulating triple-word numbers. Our new algorithms are faster than what one would obtain by just using the usual floating-point expansion algorithms in the special case of expansions of length 3, for a comparable accuracy.

#### Error analysis of some operations involved in the Fast Fourier Transform

In [44], we are interested in obtaining error bounds for the classical FFT algorithm in floating-point arithmetic, for the 2-norm as well as for the infinity norm. For that purpose we also give some results on the relative error of the complex multiplication by a root of unity, and on the largest value that can take the real or imaginary part of one term of the FFT of a vector $x$, assuming that all terms of $x$ have real and imaginary parts less than some value $b$.