On the Approximate Numerical Computation of SQRT(z) and 1/SQRT(z)

F. Pisoni and A. Castelnuovo (Italy)


Rendering, geometric algorithms, interpolation, simulation, digital signal processing.


Modern 3D-graphics applications make extensive use of square-root and inverse square-root operators to compute norms and vector normalizations. The numerical precision of these computations is not fundamental because normally, the results are eventually rounded to some fixed-point representation and many LSB(s) are dropped in this process. On the contrary, slight improvements in algorithms performances are immediately amplified, by their massive use, into tangible increases of the execution speed for the final application and extended battery life for wireless devices. The continuous demand for high peak performance and low power consumption in 3D graphics has boosted the research of new approximated methods for the numerical computation of the functions sqrt(z) and 1/sqrt(z). In this article, we give a survey on some state-of-the-art methods and introduce the use of rational Pade' approximants as an efficient alternative. Various other approximation techniques are compared, including Newton-Raphson, Lagrange and multipoint Pade'. In all these cases, the binary representation of floating-point numbers, adopted by the arithmetic unit, play a key role in the design of fast implementations for digital processors.

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