New scalar and vector elementary functions for the IBM system/370
IBM Journal of Research and Development
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The Need for an Industry Standard of Accuracy for Elementary-Function Programs
ACM Transactions on Mathematical Software (TOMS)
Computing Elementary Functions: A New Approach for Achieving High Accuracy and Good Performance
Proceedings of the Symposium on Accurate Scientific Computations
Fast evaluation of elementary mathematical functions with correctly rounded last bit
ACM Transactions on Mathematical Software (TOMS)
Hardware Starting Approximation Method and Its Application to the Square Root Operation
IEEE Transactions on Computers
Toward Correctly Rounded Transcendentals
IEEE Transactions on Computers
Fast Hardware-Based Algorithms for Elementary Function Computations Using Rectangular Multipliers
IEEE Transactions on Computers
Hardware Designs for Exactly Rounded Elementary Functions
IEEE Transactions on Computers
Fast Evaluation of the Elementary Functions in Single Precision
IEEE Transactions on Computers
A Continued-Fraction Analysis Of Trigonometric Argument Reduction
IEEE Transactions on Computers
Algorithm and Architecture for Logarithm, Exponential, and Powering Computation
IEEE Transactions on Computers
Automating custom-precision function evaluation for embedded processors
Proceedings of the 2005 international conference on Compilers, architectures and synthesis for embedded systems
Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
Reliable Implementation of Real Number Algorithms: Theory and Practice
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Unified Tables for Exponential and Logarithm Families
ACM Transactions on Mathematical Software (TOMS)
A dynamic program analysis to find floating-point accuracy problems
Proceedings of the 33rd ACM SIGPLAN conference on Programming Language Design and Implementation
Mathematical model of stored logic based computation
Mathematical and Computer Modelling: An International Journal
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The algorithms used by the IBM Israel Scientific Center for the elementary mathematical library using the IEEE standard for binary floating point arithmetic are described. The algorithms are based on the “accurate tables method.” This methodology achieves high performance and produces very accurate results. It overcomes one of the main problems encountered in elementary mathematical functions computations: achieving last bit accuracy. The results obtained are correctly rounded for almost all arguement values. Our main idea in the accurate tables method is to use “nonstandard tables,” which are different from the natural tables of equally spaced points in which the rounding error prevents obtaining last bit accuracy. In order to achieve a small error we use the following idea: Perturb the original, equally spaced, points in such a way that the table value (or tables values in case we need several tables) will be very close to numbers which can be exactly represented by the computer (much closer than the usual double percision representation). Thus we were able to control the error introduced by the computer representation of real numbers and extended the accuracy without actually using extended precision arithmetic.