Table-driven implementation of the logarithm function in IEEE floating-point arithmetic
ACM Transactions on Mathematical Software (TOMS)
Scanning polyhedra with DO loops
PPOPP '91 Proceedings of the third ACM SIGPLAN symposium on Principles and practice of parallel programming
Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic
ACM Transactions on Mathematical Software (TOMS)
Table-driven implementation of the exponential function in IEEE floating-point arithmetic
ACM Transactions on Mathematical Software (TOMS)
On integer Chebyshev polynomials
Mathematics of Computation
Elementary functions: algorithms and implementation
Elementary functions: algorithms and implementation
Computer Approximations
New Algorithms for Improved Transcendental Functions on IA-64
ARITH '99 Proceedings of the 14th IEEE Symposium on Computer Arithmetic
Scientific Computing on Itanium-Based Systems
Scientific Computing on Itanium-Based Systems
Faithful Powering Computation Using Table Look-Up and a Fused Accumulation Tree
ARITH '01 Proceedings of the 15th IEEE Symposium on Computer Arithmetic
High-Performance Architectures for Elementary Function Generation
ARITH '01 Proceedings of the 15th IEEE Symposium on Computer Arithmetic
High-performance hardware operators for polynomial evaluation
International Journal of High Performance Systems Architecture
Design issues and implementations for floating-point divide-add fused
IEEE Transactions on Circuits and Systems II: Express Briefs
Unified Tables for Exponential and Logarithm Families
ACM Transactions on Mathematical Software (TOMS)
Function approximation based on estimated arithmetic operators
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
SpringSim '10 Proceedings of the 2010 Spring Simulation Multiconference
Rigorous polynomial approximation using taylor models in Coq
NFM'12 Proceedings of the 4th international conference on NASA Formal Methods
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Polynomial approximations are almost always used when implementing functions on a computing system. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite---and sometimes small---number of bits. This is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-i coefficient has at most mi fractional bits; in other words, it is a rational number with denominator 2mi. We provide a general and efficient method for finding the best polynomial approximation under this constraint. Moreover, our method also applies if some other constraints (such as requiring some coefficients to be equal to some predefined constants or minimizing relative error instead of absolute error) are required.