Joris van der Hoeven's talk presents novel algorithms operating on formal power series. Newton's method reduces several operations to elementary computations.
Exponentiation reduces to logarithm by Newton's method.
Here, we regard the formal power series not as a list of coefficients given once and for all, but as a flow of coefficients. It is precisely this drawback which the relaxed approach solves. As all operations considered ultimately reduce to multiplications, it is enough to detail the relaxed approach for the multiplication of formal power series. Then stage 2 has to compute f 0 g 2 and f 2 g 0 only, thus a gain of one multiplication compared to the lazy approach. More generally, any relaxed algorithm for the multiplication of formal power series of order n consists of a tiling of the triangle of side n by a set of squares.