Method to compute an approximation to the reciprocal of the square root of a floating point number in IEEE format

Abstract

Approximations of reciprocal square roots are provided in IEEE floating point binary format by obtaining an index from an input value, accessing a pair of table values and performing a limited number of simple and rapidly performed manipulations. The maximum relative error in the approximation thus provided is less than 0.75/2 (2k+1) as compared with a maximum relative error of ½ k+2 of known methods, where 2 k is the number of table entries.

1 . A method for computing an approximation of a reciprocal square root, reciprocal fourth root or square root of a binary floating point number, comprising steps of 
 extracting an index comprising selected bits of said binary floating point number considered as a binary integer,    deriving a binary integer and a binary floating point number from a table in accordance with said index,    performing a shift operation on said binary floating point number considered as a binary integer to form a shifted binary integer,    subtracting said shifted binary integer from said binary floating point number derived from said table in said deriving step, and    multiplying said binary floating point number derived in said deriving step by a result of said subtracting step to provide said approximation.    
   
   
       2 . A method as recited in  claim 1 , wherein said shifting operation is a right shift by one bit.  
   
   
       3 . A method as recited in  claim 2 , wherein said extracting step comprises extracting one exponent bit and k−1 fraction bits from said binary floating point number considered as a binary integer.  
   
   
       4 . A method as recited in  claim 1 , wherein said shifting operation is a right shift by two bits.  
   
   
       5 . A method as recited in  claim 4 , wherein said extracting step comprises extracting two exponent bits and k−2 fraction bits from said binary floating point number considered as a binary integer.