Summary: We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator \(A\), a time \(t>0\), an arbitrary initial vector \(u_0\), and an error tolerance \(\epsilon>0\), the algorithm computes \(\exp(tA)u_0\) with error bounded by \(\epsilon\). The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain \(L^2(\mathbb{R}^d)\) that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like \(\exp(-cN/\log(N))\) for \(N\) quadrature points, that remains stable as \(N\rightarrow\infty\), and which is also suitable for infinite-dimensional operators. Numerical examples are given, including Schrödinger and wave equations on the aperiodic Ammann-Beenker tiling, complex perturbed fractional diffusion equations on \(L^2(\mathbb{R})\), and damped Euler-Bernoulli beam equations.