Less memory and high accuracy logarithmic number system architecture for arithmetic operations

<p>Interpolation is another important procedure for logarithmic number system (LNS) addition and subtraction. As a medium of approximation, the interpolation procedure has an urgent need to be enhanced to increase the accuracy of the operation results. Previously, most of the interpolation procedures utilized the first degree interpolators with special error correction procedure which aim to eliminate additional embedded multiplications. However, the interpolation procedure for this research was elevated up to a second degree interpolation. Proper design process, investigation, and analysis were done for these interpolation configurations in positive region by standardizing the same co-transformation procedure, which is the extended range, second order co-transformation. Newton divided differences turned out to be the best interpolator for second degree implementation of LNS addition and subtraction, with the best-achieved BTFP rate of +0.4514 and reduction of memory consumption compared to the same arithmetic used in european logarithmic microprocessor (ELM) up to 51%.</p>

The Multiplication Method with Scaling the Result for High-Precision Residue Positional Interval Logarithmic Computations

Introduction. The solution of the simulation problems critical to rounding errors, including the problems of computational mathematics, mathematical physics, optimal control, biochemistry, quantum mechanics, mathematical programming and cryptography, requires the accuracy from 100 to 1 000 decimal digits and more. The main lack of high-precision software libraries is a significant decrease of the speed-in-action, unacceptable for practical problems, in particular, when performing multiplication. A way to increase computation performance over very long numbers is using the residue number system. In this work, we discuss a new fast multiplication method with scaling the result using original hybrid residue positional interval logarithmic floating-point number representation. Materials and Methods. The new way of the organizing numerical information is a residue positional interval logarithmic number representation in which the mantissa is presented in the residue number system, and information on an absolute value (the characteristic) in the interval logarithmic number system that makes it possible to accelerate performance of comparison and scaling is developed to increase the speed of calculations; to compare modular numbers, the provisions of interval analysis are used; to scale modular numbers, the properties of the logarithmic number system and approximate interval calculations using the Chinese reminder theorem are used. Results. A new fast multiplication method of floating-point residue-represented numbers is developed and patented; the authors evaluated the developed method speed-in action, compared the developed method with classical and pipelined multiplication methods of long numbers. Discussion and Conclusion. The developed method is 2.4–4.0 times faster than the pipelined multiplication method, and is 6.4–12.9 times faster than classical multiplication methods.