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This paper is the result of concerted efforts to break the barrier between numerical accuracy and run-time efficiency in computing the fundamental decomposition of numerical linear algebra—the singular value decomposition (SVD) of general dense matrices. It is an unfortunate fact that the numerically most accurate one-sided Jacobi SVD algorithm is several times slower than generally less accurate bidiagonalization-based methods such as the QR or the divide-and-conquer algorithm. Our quest for a highly accurate and efficient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm. The new algorithm has inherited all good high accuracy properties of the Jacobi algorithm, and it can outperform the QR algorithm.