Method and apparatus for performing mathematical functions using polynomial approximation and a rectangular aspect ratio multiplier

Abstract

A method for approximating mathematical functions using polynomial expansions is implemented in a numeric processing system (10) which comprises a control and timing circuit (18), a microprogram store (20) and a multiplier circuit (34). The multiplier circuit (34) may comprise a rectangular aspect ratio multiplier circuit (40) having an additional ADDER INPUT to enable the repeated evaluation of first order polynomials to evaluate polynomial expansions associated with each mathematical function. A constant store (28) is used to store predetermined coefficients for the polynomial expansion associated with each mathematical functionsfunction. The microprogram store (20) is used to store argument transformation routines, polynomial expansions and result transformation routines associated with each mathematical function. The questions raised in reexamination request No. 90 / 004,138, filed Feb. 12, 1996, have been considered and the results thereof are reflected in this reissue patent which constitutes the reexamination certificate required by 35 U.S.C. 307 as provided in 37 CFR 1.570(e).

Description

TECHNICAL FIELD OF THE INVENTION[0001]This invention relates in general to the field of performing mathematical functions using electronic devices. More specifically, the present invention relates to a method and apparatus for performing mathematical functions using polynomial approximations in a system comprising a rectangular aspect ratio multiplier circuit.BACKGROUND OF THE INVENTION[0002]Computation of elementary and transcendental mathematical functions such as sine, cosine, logarithms and others is a required function in modern computing systems. These functions may be evaluated for any point in their domain by any of several methods. Best known among these methods are the Taylor series expansion, the Chebyshev series expansion, the CORDIC method and derivatives, Brigg's method for logarithms, Newton's method and polynomial approximation. These methods vary principally in the primitive operations they require, such as addition,and multiplication and factorial evaluation , and ...

AI Technical Summary

Benefits of technology

[0011]An important technical advantage of the present invention inheres in the fact that it uses a rectangular aspect ratio multiplier. The use of the rectangular aspect ratio multiplier saves time during several steps in the function approximation process. The operations of multiplication, division and square root involved in the transformation and polynomial evaluation processes are performed quickly through the use of new methods associated with the rectangular aspect ratio multiplier. Additionally, the short by longfull multiplier can perform full precision multiplication operations with less than full by full multiplies, depending on the number of significant bits in the operands, thus saving time and the space ordinarily required to implement a full precision multiplier.

[0012]A further technical advantage of the present invention inheres in the fact that it uses fewer constants than other approximation methods to achieve a given level of accuracy. Thus, fewer iterations are necessary to evaluate the polynomial approximation to the function and less constant storage space is needed.

[0013]Another technical advantage of the present invention inheres in the fact that the approximation to the function preserves monotonic behavior of the function. The invention thus overcomes objections of non-monotonicity frequently made regarding the use of polynomial methods of function approximation.

[0014]A final technical advantage of the present invention inheres in the fact that the constants, the arguments to the function, as well as the function are scaled. This scaling allows for a less complex multiplier to be used in the evaluation of the approximation. This scaling also allows for full precision multiplies between the constants and the arguments to be performed in less clock cycles than would be necessary for unscaled constants and arguments.

Problems solved by technology

The disadvantages of the CORDIC method are: (1) the large number of constants required to achieve a given level of accuracy, usually one for every two bits of precision in the result, (2) the large number of iterations required to produce a result, one for each constant, (3) the large number of primitive operations per iteration, usually three per iteration, and (4) the rapid accumulation of round-off error in the result, usually one unit in the last place per iteration.

The circuit complexity of the array multiplier is further complicated or, equivalently, the evaluation speed correspondingly reduced by the requirement of these other algorithms to perform full-precision multiplication of the argument by itself or by a constant.

Other disadvantages attendant to the methods other than the CORDIC method include the need for many iterations, slow or non-uniform convergence to the result, oscillatory behavior around the infinitely precise values resulting in non-monotonic behavior of the approximateapproximated function, and the requirement for additional primitive operations such as division and factorials .

The consequence of these disadvantages is larger circuit size and complexity, slower evaluation of the desired function and degraded accuracy.

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