  We present HPQCD's improved scalar, vector and tensor form factors for $B \to
K$ semileptonic decays, using the heavy-HISQ formalism for more accurate
normalisation of the weak currents. Working with masses close to the physical
$b$ on the finest ensemble and including three ensembles with physical light
quarks, we cover the full physical $q^2$ range with good precision. Our
uncertainties at $q^2=0$ are a factor of three better than earlier work.
  We compare Standard Model observables using our form factors to experimental
measurements for the rare flavour changing neutral current processes $B \to K
\ell^+\ell^-$ and $B \to K \nu\bar{\nu}$ and discuss the significance of the
tensions that arise.
