This paper presents architecture and a synthesis method for programmable numerical function generators (NFGs) for trigonometric, logarithmic, square root, and reciprocal functions. Our NFG partitions a given domain of the function into nonuniform segments using a LUT cascade, and approximates the given function by a quadratic polynomial for each segment. Thus, we can implement fast and compact NFGs for a wide range of functions. Implementation results on an FPGA show that: 1) our NFGs require only 4% of the memory needed by NFGs based on the linear approximation with nonuniform segmentation; and 2) our NFGs require only 22% of the memory needed by NFGs based on the 5th-order approximation with uniform segmentation. Our automatic synthesis system generates such compact NFGs quickly.