In this paper, we report on very efficient algorithms for spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library, which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre-associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Although the complexity of the algorithms implemented in SHTns are in (where N is the maximum harmonic degree of the transform), they perform much better than any third-party implementation, including lower-complexity algorithms, even for truncations as high as N = 1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.