Abstract
Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton–Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.
Highlights
Approximations of elementary functions are crucial in scientific computing, computer graphics, signal processing, and other fields of engineering and science [7,8,9,10]
InvSqrt1) of the fast inverse square root. It will be developed and generalized where we will show how to increase the accuracy of the InvSqrt code without losing its advantages, including the low computational cost
We focus on the Newton–Raphson corrections, which form the second part of the InvSqrt code
Summary
The main idea of the algorithm InvSqrt consists in interpreting bits of the input floating-point number as an integer [31]. In [31], we have shown that if e R = 12 ( B − 1) (e.g., e R = 63 in the 32-bit case), the function (3) (defined on integers) is equivalent to the following piece-wise linear function (when interpreted in terms of corresponding floating-point numbers):. Following and developing ideas presented in our recent papers [31,32], we propose modifications of the Newton–Raphson formulas, which result in algorithms that have the same or similar computational cost as/to InvSqrt, but improve the accuracy of the original code, even by several times.
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