Appeal No. 2004-0115 Page 2
Application No. 09/363,637
reciprocal square root. More specifically, the Newton-Raphson algorithm for
approximating the reciprocal square root of N is expressed as follows:
Xi+1 = (3 - N*Xi *Xi)*Xi/2,
where Xi is an approximation of the reciprocal square root of N at the ith iteration, \ is
greater than or equal to 1, and Xi+1 is a more accurate approximation. (Id. at 3.)
According to the appellants' method for approximating the reciprocal square root
of a number (N), a reciprocal square root of N is estimated as Xi. The estimate and N
are multiplied to produce a first intermediate result (IR1). A second intermediate
result (IR2) is determined according to the equation: IR2 = (1-Xi*IR1)/2. The second
intermediate result and the estimate are multiplied to produce a third intermediate result.
The third intermediate result and the estimate are added to produce an approximation of
the reciprocal square root of the number. (Id. at 4.)
According to the appellants, the order in which their method performs
multiplication "makes it likely that all results will be in normalized form." (Id.) Although
the Newton-Raphson algorithm requires one to determine the product of: N*Xi*Xi, they
explain that their method first multiplies N by Xi to produce an intermediate result, which
will likely be in normalized form. Their method then multiplies the intermediate result by
Xi to produce: (N*Xi)*Xi. If their method first multiplied Xi and Xi to produce the
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