We demonstrate Spiral, a domain-specific library generation system. Spiral generates high performance source code for linear transforms (such as the discrete Fourier transform and many others) directly from a problem specification. The key idea underlying Spiral is to perform automatic reasoning and optimizations at a high abstraction level using the mathematical, declarative domainspecific languages SPL and P -SPL and a rigorous rewriting framework. Optimization includes various forms of parallelization. Even though Spiral provides complete automation, its generated libraries run often faster than any existing hand-written code. Key words: Linear transform, discrete Fourier transform, FFT, domain-specific language, program generation, rewriting, matrix algebra, automatic performance tuning, multithreading, SIMD vector instructions