# Patent

## Title
Arithmetic processor utilizing multi-table look up to obtain reciprocal operands

## Abstract

Methods for determining the square root, reciprocal square root, or reciprocal of a number performed by a processor of a computer system. The methods produce high precision estimates without using iterative steps. In addition, the methods taught herein utilize compressed tables for the coefficient terms A, B, and C from the quadratic expression Ax2+Bx+C, thus minimizing hardware requirements.

## Background
BACKGROUND
1. Field of the Disclosure
The present disclosure generally relates to computer systems, and more particularly to a computer system providing results of arithmetic operations.
2. Description of the Related Art
A computer processor performs arithmetic operations on different types of numbers, or operands. For example, the simplest operations involve integer operands, which are represented using a “fixed-point” notation. Non-integers are typically represented according to a “floating-point” notation.
Many processors handle floating-point operations within a floating-point unit (FPU). Floating-point processing typically includes addition, multiplication and division operations, may also include other special mathematical operations on a single operand, such as the square root (√{square root over (x)}), reciprocal square root (1/√{square root over (x)}), and reciprocal (1/x) representing functions.
Floating point units (and other arithmetic processors) commonly use multiplier based algorithms for division. These division algorithms initially employ a seed reciprocal of the divisor provided by a lookup table system.
The seed reciprocals have a selected number of bits of accuracy. Iterative multiplies are performed to iteratively increase the accuracy of the reciprocal approximation allowing a final quotient value of predetermined accuracy to be obtained.
The seed reciprocals are typically obtained from a ROM reciprocal look-up table, or equivalent PLA (programmed logic array). The number of table input index bits and table output bits of the seed reciprocals determines the size of the look-up table. More input bits allowing more bits of accuracy in the seed reciprocals reduces the necessary number of iterative multiply cycles, reducing division time, albeit at the cost of exponential growth in the reciprocal table size.
It will be appreciated that a floating point system or method that reduces the needed number of index bits and that reduces or eliminates the need for iterative cycles in resolving operations such as reciprocal, square root, and reciprocal square root would be useful.



BRIEF DESCRIPTION OF THE DRAWINGS
The present disclosure may be better understood, and its numerous features and advantages made apparent to those skilled in the art by referencing the accompanying drawings.
FIG. 1 illustrates a block diagram of one embodiment of a processor in which an embodiment of the present disclosure may be implemented;
FIG. 2 illustrates a block diagram of a portion of a floating point unit (FPU) according to an embodiment of the present disclosure;
FIG. 3 illustrates a specific alignment of terms according to one embodiment of the disclosure; and
FIG. 4 illustrates a graph of the function |2f−13·2−27 and the odd and terminal term error term accumulation used in establishing table values for an embodiment of the present disclosure.



The use of the same reference symbols in different drawings indicates similar or identical items.
DESCRIPTION OF THE PREFERRED EMBODIMENT(S)
The present disclosure is directed to a method for evaluating arithmetic expressions, such as the square root, reciprocal square root, or reciprocal of a number, performed by a processor of a computer system. The processor evaluates the expressions using the quadratic equation

Ax2+Bx+C,

where the coefficient terms A, B, and C, are supplied by three coefficient lookup tables responsive to a common input as disclosed herein. The method results in high precision estimates without the use of iterative steps. In addition, the method taught herein uses compressed, or smaller, tables for the coefficient terms A, B, and C, thus minimizing the hardware requirements.
FIG. 1 illustrates a block diagram of one embodiment of a processor 100 in which an embodiment of the present disclosure may be implemented. While the present invention may be understood with reference to FIG. 1, this reference should not be construed in a limiting sense.
Processor 100 includes a bus interface unit 102 that controls the flow of data between processor 100 and the remainder of the data-processing system (not shown). Bus interface unit 102 is connected to both a data cache 104 and an instruction cache 106 for purposes of illustration. Instruction cache 106 supplies instructions to branch and dispatch unit 108. Branch and dispatch unit 108 determines the sequence of instructions based on current data locations and the availability of load/store unit 112, fixed-point execution unit 114, and floating-point execution unit 116, and the nature of the instructions themselves. Branch and dispatch unit 108 issues the individual instructions to the appropriate logic unit (i.e. load/store unit 112, fixed-point execution unit 114, and floating-point execution unit 116), which in turn function together to implement request operations.
Those skilled in the art will appreciate that the details of either the construction of processor 100 or its operation may vary depending on the objectives influencing the design. For example, processor 100 may include register renaming multiple fixed-point execution units 114 for executing fixed-point instructions without data dependencies in parallel, or a memory management unit regulating the content of data cache 104 and a second level (L2) cache (not shown) outside processor 100.
In one embodiment, floating-point execution unit 116 comprises selecting three input operands (A, B, and C) from tables to allow for the implementation of floating point rounded monotonic quadratic functions based on a monotonic operation. For example, the tables can provide values to support providing a floating point rounded monotonic reciprocal function. FIG. 2 illustrates a specific pipelined architecture for implementing, accessing, and using values stored in such tables. Information regarding generation of the tables to be stored in FIG. 3 to support floating point rounded monotonic functions is described subsequent to FIG. 2.
FIG. 2 illustrates a block diagram of a portion of a high precision floating point unit (FPU) according to an embodiment of the present disclosure. The most significant bits of an input operand 202, the eight most significant bits for example, are used to index a seed read only memory (ROM) 204. It will be appreciated that operation information can also be used to index seed ROM 204. Operation information can include identification of various of operations, such as reciprocal or square root reciprocal. ROM 204 is part of a table look-up system that stores multiple component tables. For example, the ROM 204 can include tables to provide operands A, B, and C to be stored in registers 206, 208, and 210, respectively. The generation of the values contained in seed ROM 204 is discussed below.
One advantage of the implementation described herein is that the tables stored within ROM 204 can be accessed using a single index value having one-third the number of bits of the floating point result generated. For example, for a 24-bit result accurate to a Unit in the Last Place (ULP), only an eight-bit index is needed. The ability to provide this level of precision in the result using a ROM table of this size is advantageous.
Mid-range bits of input operand 202 are used to determine a nine-bit input to the square circuit 212, which may be a ROM or comprise a small multiplier, and a Booth recoding. Inverter 214 and multiplexer 216 are utilized to reduce the size of square circuit 212 by substantially one half, which is discussed in greater detail later in this disclosure. The operand output of square circuit 212 is stored in Booth recoded format in register 218 for input to a multiplier.
The least significant bits of input operand 202 are input to Booth recoder 220. The output of Booth recoder 220 is stored in register 222 for input to a multiplier.
Note that in the specific implementation illustrated, that the ROM table 204 is indexed by a portion of bits of the input operand that represent the operand's higher order bits. In addition, the ROM 204 is coupled to receive index bits that are mutually exclusive to the bits used to access the squaring circuit 212 and Booth recoder 220. However, the square circuit 212 and Booth recoder 220 are coupled to receive common, or overlapping, bits. While the portions of the input operand received at the ROM 212, ROM 204 and recoder 220 are illustrated as being adjacent bit sequences, they may be non-adjacent bit sequences in other embodiments.
Operand A from register 206 and the operand from register 218 are multiplied by multiplier 224. Multiplier 224 can be a carry save multiplier with redundant outputs and no ripple carry. Operand B from register 208 and the operand from register 222 are multiplied by multiplier 226. Multiplier 226 can also be a carry save multiplier with redundant outputs and no ripple carry. The redundant carry save outputs of multipliers 224 and 226 are added by four-to-two adder 228 producing a redundant carry save representation of the Ax2+Bx term with the carry and sum values stored in registers 230 and 232, respectively. Operand C from register 210 is stored in register 234 to maintain a parallel pipeline. An implicit bit is appended to operand C from register 210 by appending implicit bit unit 236 forming operand C′. The carry and sum operands of the Ax2+Bx term are added together with operand C′ by three-to-one adder 238. The output of adder 238 is normalized into the format specified for the floating point output, and the resulting reciprocal value is stored in reciprocal register 242.
FIG. 3 is a block diagram illustrating an alignment of terms according to at least one embodiment of the present disclosure. During the table lookup and recoding stage of operation, the 8 most significant bits (79:72) of the input operand 203, excluding the implicit bit 80, are used as the index into the table of ROM 204. Bits 9 through 23 (71:57) of the input operand are used in calculation of the Bx term, and bits 9 through 18 (71:62) are used to access ROM 212 to calculate the x2 component. A further step is used to permit the squaring circuit to be reduced in size. By centering the effect of the squared term, the size of the table used to calculate the square can be reduced by half by using what was formerly the most significant bit to invert the index into table. Since the square input is used as a multiplier input, the data that is stored in a ROM implementation of the squaring circuit can be precomputed recoded to the format expected by the multiplier, thus removing a Booth recoder from the path. The x input to the Bx term is recoded in this stage.
In the multiply or second stage, both products are calculated in carry save multipliers. The results for both products are added together in a 4 to 2 adder, and the redundant result is passed to the add stage.
In stage three, or the add stage, an implicit bit is appended to the front of the C term. The two parts of the redundant number from the multiply stage are added with the C term to form the result. The values stored in the seed ROM are chosen to provide a rounded function as the result, when truncated to the output precision size during normalization. The rounded result is normalized in normalize block 240. For the reciprocal, square root reciprocal, and square root functions, the normalization comprises forcing the output to an exact power of two if a carry out is detected from the adder stage. The normalize unit also removes the implicit bit before storing the result in the output register 242.
The tables stored in ROM 204 are chosen to allow for implementing a floating point rounded monotonic quadratic function based on a monotonic operation. Specifics of generating the tables of ROM 204 are described below.
Table Compression
A monotonic operation applied to a single operand is a monotonic function of that operand. This is the case for the monotonic operations reciprocal, square root, and square root reciprocal. A monotonic function, as the term monotonic implies, is always changing in the same direction. A monotonic increasing function of a variable y increases or stays constant as y increases, but never decreases. A monotonic decreasing function of y decreases or stays constant as y increases, but never increases. Each term in a monotonic series is, in an increasing monotonic, greater than or equal to the one before it. Or, in a decreasing monotonic, less than or equal to the one before it.
A floating point rounded function over an interval has a discrete set of floating point inputs over that interval. The value of the floating point rounded function over the interval is a step function, with the steps decreasing or remaining the same as the floating point inputs increase for a decreasing monotonic floating point rounded function, and with the steps increasing or remaining the same as the floating point inputs increase for an increasing monotonic function.
It is sufficient for the reciprocal operation to consider floating point inputs over the interval [1, 2). Thus for example y=1.b1b2 . . . b23 provides the 223 (about 8 million) input values for the 24 bit precision specified in the IEEE standard single precision format. For the square root and square root reciprocal operations it is necessary to consider two input binades covering [1, 4).
For evaluation of a floating point rounded polynomial function, such as the quadratic A y2+By+C, where all coefficients A, B, C are determined from a leading bit portion such as y8=1.b1b2 . . . b8 of y, the floating point rounded quadratic function is then a piecewise quadratic function, here having 28=256 pieces over [1, 2).
The verification that such a floating point rounded quadratic function is monotonic can be verified by observing that each piece is monotonic and by verifying that the joining points of the pieces (here 255 in number) preserver monotonicity. This test confirming monotonicity as well as tests for the whole single precision range of 8-16 million cases for each reciprocal, reciprocal square root and square root reciprocal are easy to perform in addition to theoretical arguments from the level of accuracy and the derivatives of the function.
The simplicity of these tests allows perturbation of the coefficients to improve the percent round to nearest results or to reduce table size without losing the monotonicity or unit in the last place accuracy of the floating point rounded function realizing the monotonic operation.
The following paragraphs describe the calculations to provide an 8-bit y table solution for table compression such that a plurality of tables, three in our example, are indexed by an index value three times smaller bitwise than the floating point result, i.e., 8-bit tables produce a 24 bit floating point result.
Table Compression (8-bit y Table Solution)
Let




  
    
      
        
          y
          =
          
            
              
                
                  1.
                  ⁢
                  
                    b
                    1
                  
                  ⁢
                  
                    b
                    
                      2
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  ...
                  ⁢
                  
                    
                  
                  ⁢
                  
                    b
                    8
                  
                
                
                  y
                  8
                
              
              ⁢
              
                
                  
                    b
                    9
                  
                  ⁢
                  
                    b
                    10
                  
                  ⁢
                  
                    
                  
                  ⁢
                  ...
                  ⁢
                  
                    
                  
                  ⁢
                  
                    b
                    23
                  
                
                f
              
            
            =
            
              
                y
                8
              
              +
              
                f
                ⁢
                
                  
                
                ⁢
                
                  2
                  
                    -
                    8
                  
                
              
            
          
        
        ,
        
          
        
        ⁢
        where
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        
          
            y
            8
          
          =
          
            1.
            ⁢
            
              b
              1
            
            ⁢
            
              b
              2
            
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              
            
            ⁢
            
              b
              8
            
          
        
        ,
        
          
        
        ⁢
        and
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          f
          =
          
            0.
            ⁢
            
              b
              9
            
            ⁢
            
              b
              10
            
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              
            
            ⁢
            
              
                b
                23
              
              ⁡
              
                (
                
                  15
                  ⁢
                  
                    
                  
                  ⁢
                  bits
                
                )
              
            
          
        
        ,
        
          
        
        ⁢
        
          
            where
            ⁢
            
              
            
            ⁢
            0
          
          ⁢
          
            <
            _
          
          ⁢
          f
          ⁢
          
            <
            _
          
          ⁢
          
            1
            -
            
              
                2
                
                  -
                  15
                
              
              .
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




The midpoint is then given by

y8+2−9=1.b1b2.b8,  (4)

and the center point by

(2f−1)=(b9b10 . . . b23)−1, where −1≦(2f−1)≦1−2−14.  (5)

Thus the expression for y is given by

y=(y8+2−9)+(2f−1)2−9.  (6)

The following identity for the reciprocal of y is given by the expression:





  
    
      
        
          1
          y
        
        =
        
          
            1
            
              
                y
                8
              
              +
              
                2
                
                  -
                  9
                
              
            
          
          -
          
            
              
                (
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    f
                  
                  -
                  1
                
                )
              
              
                
                  y
                  8
                
                +
                
                  2
                  
                    -
                    9
                  
                
              
            
            ⁢
            
              1
              y
            
            ⁢
            
              2
              
                -
                9
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




The preceding equation (7) may be checked by placing terms over a common denominator.
Iterative substitution for 1/y two times yields a cubic expression in (2f−1) as follows:




  
    
      
        
          1
          y
        
        =
        
          
            1
            
              
                
                  y
                  8
                
                +
                
                  2
                  
                    -
                    9
                  
                
              
              
                }
                A
              
            
          
          -
          
            
              
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      f
                    
                    -
                    1
                  
                  )
                
                
                  
                    (
                    
                      
                        y
                        8
                      
                      +
                      
                        2
                        
                          -
                          9
                        
                      
                    
                    )
                  
                  2
                
              
              ⁢
              
                2
                
                  -
                  9
                
              
            
            
              }
              
                
                  -
                  B
                
                ⁢
                
                  
                
                ⁢
                
                  2
                  
                    -
                    9
                  
                
              
            
          
          +
          
            
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        f
                      
                      -
                      1
                    
                    )
                  
                  2
                
                
                  
                    (
                    
                      
                        y
                        8
                      
                      +
                      
                        2
                        
                          -
                          9
                        
                      
                    
                    )
                  
                  3
                
              
              ⁢
              
                2
                
                  -
                  18
                
              
            
            
              }
              
                
                  +
                  C
                
                ⁢
                
                  
                
                ⁢
                
                  2
                  
                    -
                    18
                  
                
              
            
          
          -
          
            
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          f
                        
                        -
                        1
                      
                      )
                    
                    3
                  
                  
                    
                      (
                      
                        
                          y
                          8
                        
                        +
                        
                          2
                          
                            -
                            9
                          
                        
                      
                      )
                    
                    3
                  
                
                ⁢
                
                  (
                  
                    1
                    y
                  
                  )
                
                ⁢
                
                  2
                  
                    -
                    27
                  
                
              
              
                }
                
                  
                    -
                    D
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    2
                    
                      -
                      27
                    
                  
                
              
            
            .
          
        
      
    
    
      
        (
        8
        )
      
    
  




Note the algebraic expression 8 is an exact equation for the reciprocal, where the first three terms can be evaluated to as much accuracy as desired by a lookup table indexed by 8-bits along with appropriate multiplications. Only the fourth term, which constitutes less than 1⁄8 of a unit relative to our target precision of 24-bits, needs to be approximated.
Tables for the A, B, and C coefficients for a reciprocal and a square root reciprocal which have been generated utilizing the mathematical methodology herein are presented in the Appendix.
Our invention employs several techniques to generate the coefficients C0, C1, C2 of the rounded floating point function approx (1/y) to reduce table size and computation effort while maintaining the monotonicity property with unit in the last place (ulp) accuracy in realizing the reciprocal operation.

    
    
        The coefficient C0 includes the half ulp increment so that the final rounding occurs simply by truncating the result during normalization to the floating point output format.
        The coefficient C2 includes compensation for the rounding error in providing C0, so that the total coefficient rounding errors from C0 and C2 are correlated to provide a net rounding error no worse than the worse case of a single rounding error, as described in co pending application Ser. No. 10/108,251, filed on Mar. 26, 2002, entitled APPARATUS AND. METHOD FOR MINIMIZING ACCUMULATED ROUNDING ERRORS IN COEFFICIENT VALUES IN A LOOKUP TABLE FOR INTERPOLATING POLYNOMIALS, naming inventor David W. Matula, which is incorporated herein by reference.
        The coefficient of the linear term C1 includes a linear approximation to the cubic term of equation 8 to increase the accuracy so as to reduce the table size for the final 3 coefficients C0, C1, C2 of our floating point function approx (1/y).

Error Roundoff Methodology

    
    


If we let




  
    C
    1
  
  =
  
    
      
        
          RN
          18
        
        ⁡
        
          (
          
            
              1
              
                
                  (
                  
                    
                      y
                      8
                    
                    +
                    
                      2
                      
                        -
                        9
                      
                    
                  
                  )
                
                2
              
            
            +
            
              
                
                  169
                  ⁢
                  
                    /
                  
                  ⁢
                  225
                
                
                  
                    
                      (
                      
                        
                          y
                          8
                        
                        +
                        
                          2
                          
                            -
                            9
                          
                        
                      
                      )
                    
                    3
                  
                  ⁢
                  y
                
              
              ⁢
              
                2
                
                  -
                  18
                
              
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      and
      ⁢
      
        
      
      ⁢
      
        ρ
        1
      
      ⁢
      
        2
        
          -
          19
        
      
    
    =
    
      
        (
        
          
            1
            
              
                (
                
                  
                    y
                    8
                  
                  +
                  
                    2
                    
                      -
                      9
                    
                  
                
                )
              
              2
            
          
          +
          
            
              
                169
                ⁢
                
                  /
                
                ⁢
                225
              
              
                
                  
                    (
                    
                      
                        y
                        8
                      
                      +
                      
                        2
                        
                          -
                          9
                        
                      
                    
                    )
                  
                  3
                
                ⁢
                y
              
            
            ⁢
            
              2
              
                -
                18
              
            
          
        
        )
      
      -
      
        C
        1
      
    
  




Then.



  
    
      1
      y
    
    =
    
      
        C
        0
      
      +
      
        
          C
          2
        
        ⁢
        Q
        ⁢
        
          
        
        ⁢
        
          2
          
            -
            18
          
        
      
      +
      
        
          ρ
          α
        
        ⁢
        
          2
          
            -
            28
          
        
      
      +
      
        
          C
          2
        
        ⁢
        
          ρ
          Q
        
        ⁢
        
          2
          
            -
            28
          
        
      
      +
      
        
          
            C
            2
          
          ⁡
          
            (
            
              
                2
                ⁢
                
                  
                
                ⁢
                f
              
              -
              1
            
            )
          
        
        ⁢
        
          ρ
          ⁡
          
            (
            f
            )
          
        
        ⁢
        
          2
          
            -
            27
          
        
      
      +
      
        
          
            C
            1
          
          ⁡
          
            (
            
              
                2
                ⁢
                
                  
                
                ⁢
                f
              
              -
              1
            
            )
          
        
        ⁢
        
          ρ
          ⁡
          
            (
            f
            )
          
        
        ⁢
        
          2
          
            -
            9
          
        
      
      +
      
        
          (
          
            
              2
              ⁢
              
                
              
              ⁢
              f
            
            -
            1
          
          )
        
        ⁢
        
          ρ
          1
        
        ⁢
        
          2
          
            -
            28
          
        
      
      +
      
        
          (
          
            
              f
              _
            
            
              
                y
                _
              
              4
            
          
          )
        
        ⁢
        
          2
          
            -
            29
          
        
      
      +
      
        
          
            ∂
            ′
          
          ⁢
          
            2
            
              -
              28
            
          
        
        ⁢
        
          
        
        ⁢
        where
      
    
  




  
    
      
        
          
            
              f
              _
            
            =
            
              32
              ⁢
              
                (
                
                  f
                  -
                  
                    1
                    2
                  
                
                )
              
              ⁢
              
                (
                
                  f
                  -
                  
                    1
                    15
                  
                
                )
              
              ⁢
              
                (
                
                  f
                  -
                  
                    14
                    15
                  
                
                )
              
            
          
          ,
          
            
          
          ⁢
          
            and
            ⁢
            
              
            
            ⁢
            
              
                ∂
                ′
              
              ⁢
              is
            
            ⁢
            
              
            
            ⁢
            less
            ⁢
            
              
            
            ⁢
            than
            ⁢
            
              
            
            ⁢
            1.
          
        
      
    
    
      
        
          
            
              y
              _
            
            4
          
          =
          
            
              
                (
                
                  
                    Y
                    8
                  
                  +
                  
                    2
                    
                      -
                      9
                    
                  
                
                )
              
              3
            
            ⁢
            
              y
              .
            
          
        
      
    
  



Let us define an approximation,




  
    approx
    ⁡
    
      (
      
        1
        y
      
      )
    
  
  ,




in terms of our table lookup and multiply accumulate in finite precision by the expression:





  
    approx
    ⁡
    
      (
      
        1
        y
      
      )
    
  
  =
  
    
      C
      0
    
    +
    
      
        
          C
          1
        
        ⁡
        
          (
          
            
              2
              ⁢
              
                
              
              ⁢
              f
            
            -
            1
          
          )
        
      
      ⁢
      
        2
        
          -
          9
        
      
    
    +
    
      
        C
        2
      
      ⁢
      Q
      ⁢
      
        
      
      ⁢
      
        2
        
          -
          18
        
      
    
  




Then the error term identification is as follows:





  
    
      (
      
        1
        y
      
      )
    
    -
    
      approx
      ⁡
      
        (
        
          1
          y
        
        )
      
    
  
  =
  
    
      
        ρ
        α
      
      ⁢
      
        2
        
          -
          28
        
      
    
    +
    
      
        (
        
          
            2
            ⁢
            
              
            
            ⁢
            f
          
          -
          1
        
        )
      
      ⁢
      
        ρ
        1
      
      ⁢
      
        2
        
          -
          28
        
      
    
    -
    
      
        (
        
          
            f
            _
          
          
            
              y
              _
            
            4
          
        
        )
      
      ⁢
      
        2
        
          -
          29
        
      
    
    +
    
      
        C
        2
      
      ⁢
      
        ρ
        Q
      
      ⁢
      
        2
        
          -
          28
        
      
    
    +
    
      
        
          C
          2
        
        ⁡
        
          (
          
            
              2
              ⁢
              
                
              
              ⁢
              f
            
            -
            1
          
          )
        
      
      ⁢
      
        ρ
        ⁡
        
          (
          f
          )
        
      
      ⁢
      
        2
        
          -
          27
        
      
    
  




Where the portion of the expression ρα2−28 represents the even terms roundoff, the portion




  
    
      (
      
        
          2
          ⁢
          
            
          
          ⁢
          f
        
        -
        1
      
      )
    
    ⁢
    
      ρ
      1
    
    ⁢
    
      2
      
        -
        28
      
    
  
  -
  
    
      (
      
        
          f
          _
        
        
          
            y
            _
          
          4
        
      
      )
    
    ⁢
    
      2
      
        -
        29
      
    
  




represents the odd terms roundoff and function termination, the portion C2ρQ2−28 represents the squaring term roundoff, and the portion of the expression C2(2f−1)ρ(f)227 represents the approximation error for the square.

The behavior of the portions of the error terms expression can be understood with reference to Table 1.







TABLE 1







Behavior of Error Terms










ρα2−28




  
    
      (
      
        
          2
          ⁢
          
            
          
          ⁢
          f
        
        -
        1
      
      )
    
    ⁢
    
      ρ
      1
    
    ⁢
    
      2
      
        -
        28
      
    
  
  -
  
    
      (
      
        
          f
          _
        
        
          
            y
            _
          
          4
        
      
      )
    
    ⁢
    
      2
      
        -
        29
      
    
  




C2ρQ2−28
C2(2f − 1)ρ(f)2−27





Essentiallyuniformover(−1,1) ·2−28,indepen-dent ofy and f.
Sum can be correlated to bebounded by



  
    5
    4
  
  ⁢
  
    
      2
      
        -
        28
      
    
    .
  



Non linear term diminishesrapidly with y→2.
Essentially



  
    
      ρ
      Q
    
    
      y
      3
    
  
  ⁢
  
    
      2
      
        -
        28
      
    
    .
  



Dampsrapidlywith y→2.
Essentially



  
    
      
        2
        ⁢
        
          
        
        ⁢
        f
      
      -
      1
    
    
      y
      3
    
  
  ⁢
  
    ρ
    ⁡
    
      (
      f
      )
    
  
  ⁢
  
    
      2
      
        -
        27
      
    
    .
  



Damps rapidlywith y→2 and|2f − 1|→0.









The damping of error terms can be understood with reference to Table 2.







TABLE 2







Error Term Damping












Un-

y ≧
|2f − 1| ≦



restricted
y ≧ 11⁄4
11⁄2
1⁄2



Upper
Upper
Upper
Upper


Error Term
Bound
Bound
Bound
Bound














ρα2−28
1
1
1
1


 






  
    
      (
      
        
          2
          ⁢
          
            
          
          ⁢
          f
        
        -
        1
      
      )
    
    ⁢
    
      ρ
      1
    
    ⁢
    
      2
      
        -
        28
      
    
  
  -
  
    
      (
      
        
          f
          _
        
        
          
            y
            _
          
          4
        
      
      )
    
    ⁢
    
      2
      
        -
        29
      
    
  




3/2
1.2
1.1
1


 


C2ρQ2−28
1
0.5
0.3
1


C2(2f − 1)ρ(f)2−27
2
1
0.6
1


Independent Summation
5.5
3.7
3
4


Bound (units)
(×2−28)
(×2−28)
(×2−28)
(×2−28)









It is preferable to have




  
    
    
      
        (
        
          1
          y
        
        )
      
      -
      
        approx
        ⁡
        
          (
          
            1
            y
          
          )
        
      
    
    
  
  <
  
    3.5
    ·
    
      
        2
        
          -
          28
        
      
      .
    
  




The difficult cases should be found in the restricted range 1≦y≦ 5/4, and |2f−1|≧1⁄2. The worst case compounding of all four errors above is unlikely given the relatively small number of cases. It should be noted that there are only about 64 values for ρα and ρ1 over the range 1≦y≦11⁄4, and only about 256 values for ρQ over |2f−1|≧1⁄2. Perturbation of coefficients by one, at most, unit in the last place (ULP) each can also avoid worst cases.

An advantage of the present disclosure for the use of floating point rounding to realize a monotonic operation to construct the coefficient tables shown in Appendix A is that the higher order (cubic) term approximation can be incorporated into the linear term such that no fourth coefficient term table is needed to realize a higher than quadratic level approximation. This concept is illustrated in FIG. 4, which illustrates a graph of the function |2f−1|32−27 and the odd and terminal term error term accumulation in which the present disclosure may be implemented. The functions graphed are sign symmetric about f=1⁄2.
The above-disclosed subject matter is to be considered illustrative, and not restrictive, and the appended claims are intended to cover all such modifications, enhancements, and other embodiments that fall within the true spirit and scope of the present invention. Thus, to the maximum extent allowed by law, the scope of the present invention is to be determined by the broadest permissible interpretation of the following claims and their equivalents, and shall not be restricted or limited by the foregoing detailed description.







APPENDIX







A—COEFFICIENT TABLE FOR RECIPROCAL









// c2 DIV



width = 10



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B—COEFFICIENT TABLE FOR RECIPROCAL









// c1 DIV



width = 19



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@0ff 20100







C—COEFFICIENT TABLE FOR RECIPROCAL









// c0 DIV



width = 28



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@0ff 804017c







A—COEFFICIENTS FOR ODD SQUARE ROOT RECIPROCAL









// c2 SQRT



odd width = 9



@000 17d



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@025 111



@026 110



@027 10d



@028 109



@029 108



@02a 105



@02b 104



@02c 102



@02d 100



@02e fd



@02f fc



@030 f9



@031 f8



@032 f5



@033 f4



@034 f0



@035 f0



@036 ed



@037 ec



@038 e9



@039 e8



@03a e5



@03b e5



@03c e1



@03d e0



@03e de



@03f dd



@040 da



@041 d8



@042 d8



@043 d5



@044 d4



@045 d2



@046 d1



@047 d0



@048 cd



@049 cc



@04a cc



@04b c9



@04c c8



@04d c6



@04e c6



@04f c4



@050 c2



@051 c1



@052 c0



@053 bd



@054 bc



@055 b9



@056 b9



@057 b8



@058 b8



@059 b5



@05a b4



@05b b4



@05c b1



@05d b0



@05e b0



@05f ad



@060 ac



@061 ac



@062 aa



@063 a8



@064 a8



@065 a8



@066 a4



@067 a4



@068 a2



@069 a2



@06a a1



@06b a0



@06c a0



@06d a0



@06e 9c



@06f 9c



@070 9a



@071 99



@072 98



@073 98



@074 96



@075 95



@076 95



@077 94



@078 92



@079 91



@07a 91



@07b 90



@07c 90



@07d 8d



@07e 8c



@07f 8c



@080 8b



@081 89



@082 89



@083 88



@084 88



@085 88



@086 86



@087 85



@088 84



@089 84



@08a 82



@08b 81



@08c 80



@08d 80



@08e 80



@08f 7d



@090 7d



@091 7c



@092 7c



@093 7c



@094 7a



@095 79



@096 78



@097 78



@098 78



@099 76



@09a 75



@09b 75



@09c 75



@09d 74



@09e 74



@09f 72



@0a0 72



@0a1 71



@0a2 70



@0a3 70



@0a4 70



@0a5 70



@0a6 6d



@0a7 6c



@0a8 6c



@0a9 6c



@0aa 6c



@0ab 6b



@0ac 69



@0ad 69



@0ae 69



@0af 69



@0b0 68



@0b1 68



@0b2 65



@0b3 65



@0b4 64



@0b5 64



@0b5 64



@0b7 64



@0b8 63



@0b9 61



@0ba 62



@0bb 61



@0bc 61



@0bd 60



@0be 60



@0bf 60



@0c0 5e



@0c1 5d



@0c2 5e



@0c3 5d



@0c4 5c



@0c5 5c



@0c6 5c



@0c7 5c



@0c8 59



@0c9 5a



@0ca 5a



@0cb 59



@0cc 58



@0cd 58



@0ce 58



@0cf 58



@0d0 56



@0d1 55



@0d2 55



@0d3 55



@0d4 54



@0d5 54



@0d5 54



@0d7 54



@0d8 52



@0d9 52



@0da 52



@0db 52



@0dc 51



@0dd 50



@0de 50



@0df 50



@0e0 50



@0e1 50



@0e2 4e



@0e3 50



@0e4 4d



@0e5 4e



@0e6 4c



@0e7 4d



@0e8 4c



@0e9 4c



@0ea 4c



@0eb 4c



@0ec 4b



@0ed 4a



@0ee 4a



@0ef 49



@0f0 49



@0f1 49



@0f2 48



@0f3 48



@0f4 48



@0f5 48



@0f6 48



@0f7 48



@0f8 48



@0f9 46



@0fa 46



@0fb 44



@0fc 45



@0fd 44



@0fe 45



@0ff 44







A—COEFFICIENTS FOR EVEN SQUARE ROOT RECIPROCAL









// c2 SQRT



even



@100 110



@101 10c



@102 108



@103 108



@104 104



@105 101



@106 100



@107 fd



@108 fa



@109 f8



@10a f5



@10b f4



@10c f1



@10d f0



@10e ec



@10f ea



@110 e9



@111 e6



@112 e4



@113 e1



@114 e0



@115 dd



@116 dc



@117 da



@118 d8



@119 d5



@11a d4



@11b d1



@11c d1



@11d d0



@11e cd



@11f cc



@120 c9



@121 c8



@122 c6



@123 c5



@124 c4



@125 c0



@126 c0



@127 be



@128 bc



@129 ba



@12a b8



@12b b8



@12c b5



@12d b5



@12e b2



@12f b1



@130 b1



@131 b0



@132 ac



@133 ac



@134 aa



@135 a8



@136 a8



@137 a5



@138 a5



@139 a4



@13a a2



@13b a0



@13c a0



@13d a0



@13e 9d



@13f 9c



@140 9c



@141 99



@142 98



@143 98



@144 96



@145 95



@146 94



@147 94



@148 92



@149 90



@14a 90



@14b 90



@14c 8d



@14d 8c



@14e 8a



@14f 89



@150 88



@151 88



@152 88



@153 88



@154 84



@155 84



@156 84



@157 82



@158 82



@159 80



@15a 80



@15b 7e



@15c 7d



@15d 7c



@15e 7c



@15f 7c



@160 79



@161 79



@162 79



@163 78



@164 78



@165 75



@166 75



@167 74



@168 74



@169 72



@16a 72



@16b 71



@16c 70



@16d 70



@16e 70



@16f 6d



@170 6d



@171 6c



@172 6c



@173 6a



@174 6a



@175 69



@176 68



@177 69



@178 68



@179 68



@17a 66



@17b 66



@17c 65



@17d 64



@17e 64



@17f 64



@180 62



@181 61



@182 60



@183 61



@184 60



@185 60



@186 60



@187 60



@188 5c



@189 5c



@18a 5c



@18b 5c



@18c 5c



@18d 59



@18e 59



@18f 58



@190 58



@191 58



@192 58



@193 58



@194 58



@195 56



@196 55



@197 54



@198 54



@199 54



@19a 54



@19b 52



@19c 52



@19d 51



@19e 51



@19f 50



@1a0 50



@1a1 50



@1a2 50



@1a3 50



@1a4 4e



@1a5 4d



@1a6 4d



@1a7 4d



@1a8 4d



@1a9 4c



@1aa 4c



@1ab 4c



@1ac 4b



@1ad 49



@1ae 4a



@1af 49



@1b0 49



@1b1 48



@1b2 49



@1b3 48



@1b4 48



@1b5 48



@1b6 48



@1b7 45



@1b8 46



@1b9 45



@1ba 45



@1bb 45



@1bc 44



@1bd 44



@1be 44



@1bf 44



@1c0 44



@1c1 41



@1c2 41



@1c3 42



@1c4 41



@1c5 40



@1c6 40



@1c7 40



@1c8 40



@1c9 40



@1ca 40



@1cb 40



@1cc 40



@1cd 3e



@1ce 3e



@1cf 3e



@1d0 3d



@1d1 3c



@1d2 3c



@1d3 3c



@1d4 3d



@1d5 3c



@1d6 3c



@1d7 3c



@1d8 3a



@1d9 3a



@1da 39



@1db 3a



@1dc 39



@1dd 39



@1de 39



@1df 39



@1e0 38



@1e1 38



@1e2 38



@1e3 38



@1e4 38



@1e5 38



@1e6 38



@1e7 36



@1e8 35



@1e9 36



@1ea 34



@1eb 35



@1ec 35



@1ed 35



@1ee 35



@1ef 34



@1f0 34



@1f1 34



@1f2 34



@1f3 34



@1f4 34



@1f5 31



@1f6 31



@1f7 32



@1f8 31



@1f9 31



@1fa 32



@1fb 31



@1fc 31



@1fd 30



@1fe 30



@1ff 30







B—COEFFICIENTS FOR ODD SQUARE ROOT RECIPROCAL









// c1 SQRT



odd width = 18



@000 3fd02



@001 3f711



@002 3f12f



@003 3eb5b



@004 3e595



@005 3dfde



@006 3da34



@007 3d498



@008 3cf0a



@009 3c989



@00a 3c416



@00b 3beaf



@00c 3b955



@00d 3b408



@00e 3aec7



@00f 3a993



@010 3a46b



@011 39f4f



@012 39a3f



@013 3953b



@014 39042



@015 38b55



@016 38673



@017 3819c



@018 37cd1



@019 37810



@01a 3735a



@01b 36eae



@01c 36a0d



@01d 36576



@01e 360ea



@01f 35c68



@020 357f0



@021 35381



@022 34f1d



@023 34ac2



@024 34670



@025 34229



@026 33dea



@027 339b4



@028 33588



@029 33164



@02a 32d4a



@02b 32938



@02c 3252e



@02d 3212e



@02e 31d36



@02f 31946



@030 3155e



@031 3117e



@032 30da7



@033 309d8



@034 30611



@035 30251



@036 2fe99



@037 2fae9



@038 2f740



@039 2f39f



@03a 2f005



@03b 2ec73



@03c 2e8e7



@03d 2e563



@03e 2e1e6



@03f 2de70



@040 2db00



@041 2d798



@042 2d436



@043 2d0db



@044 2cd87



@045 2ca39



@046 2c6f2



@047 2c3b1



@048 2c076



@049 2bd42



@04a 2ba14



@04b 2b6ec



@04c 2b3ca



@04d 2b0ae



@04e 2ad98



@04f 2aa88



@050 2a77d



@051 2a479



@052 2a17a



@053 29e81



@054 29b8d



@055 2989f



@056 295b6



@057 292d3



@058 28ff5



@059 28d1d



@05a 28a4a



@05b 2877c



@05c 284b3



@05d 281ef



@05e 27f30



@05f 27c76



@060 279c1



@061 27712



@062 27466



@063 271c0



@064 26f1e



@065 26c82



@066 269e9



@067 26756



@068 264c7



@069 2623c



@06a 25fb6



@06b 25d35



@06c 25ab8



@06d 2583f



@06e 255cb



@06f 2535a



@070 250ee



@071 24e86



@072 24c23



@073 249c3



@074 24768



@075 24510



@076 242bd



@077 2406e



@078 23e22



@079 23bdb



@07a 23997



@07b 23757



@07c 2351a



@07d 232e2



@07e 230ad



@07f 22e7c



@080 22c4f



@081 22a25



@082 227ff



@083 225dc



@084 223bd



@085 221a2



@086 21f89



@087 21d74



@088 21b63



@089 21955



@08a 2174a



@08b 21543



@08c 2133f



@08d 2113e



@08e 20f40



@08f 20d46



@090 20b4e



@091 2095a



@092 20769



@093 2057b



@094 20390



@095 201a8



@096 1ffc3



@097 1fde2



@098 1fc02



@099 1fa26



@09a 1f84d



@09b 1f677



@09c 1f4a3



@09d 1f2d3



@09e 1f105



@09f 1ef3a



@0a0 1ed71



@0a1 1ebac



@0a2 1e9e9



@0a3 1e829



@0a4 1e66b



@0a5 1e4b1



@0a6 1e2f8



@0a7 1e143



@0a8 1df90



@0a9 1dddf



@0aa 1dc31



@0ab 1da86



@0ac 1d8dd



@0ad 1d736



@0ae 1d592



@0af 1d3f0



@0b0 1d251



@0b1 1d0b4



@0b2 1cf1a



@0b3 1cd82



@0b4 1cbec



@0b5 1ca58



@0b6 1c8c7



@0b7 1c738



@0b8 1c5ac



@0b9 1c422



@0ba 1c299



@0bb 1c113



@0bc 1bf90



@0bd 1be0e



@0be 1bc8f



@0bf 1bb11



@0c0 1b996



@0c1 1b81d



@0c2 1b6a6



@0c3 1b531



@0c4 1b3bf



@0c5 1b24e



@0c6 1b0df



@0c7 1af72



@0c8 1ae08



@0c9 1ac9f



@0ca 1ab38



@0cb 1a9d3



@0cd 1a710



@0ce 1a5b0



@0cf 1a453



@0d0 1a2f8



@0d1 1a19f



@0d2 1a047



@0d3 19ef1



@0d4 19d9d



@0d5 19c4b



@0d6 19afb



@0d7 199ac



@0d8 1985f



@0d9 19714



@0da 195cb



@0db 19484



@0dc 1933e



@0dd 191fa



@0de 190b7



@0df 18f77



@0e0 18e38



@0e1 18cfa



@0e2 18bbe



@0e3 18a84



@0e4 1894c



@0e5 18815



@0e6 186df



@0e7 185ac



@0e8 1847a



@0e9 18349



@0ea 1821a



@0eb 180ec



@0ec 17fc0



@0ed 17e96



@0ee 17d6d



@0ef 17c46



@0f0 17b20



@0f1 179fb



@0f2 178d8



@0f3 177b7



@0f4 17696



@0f5 17578



@0f6 1745b



@0f7 1733f



@0f8 17224



@0f9 1710b



@0fa 16ff4



@0fb 16edd



@0fc 16dc8



@0fd 16cb5



@0fe 16ba3



@0ff 16a92







B—COEFFICENTS FOR EVEN SQUARE ROOT RECIPROCAL









// c1 SQRT even



@100 2d1f6



@101 2cdc3



@102 2c99a



@103 2c57b



@104 2c166



@105 2bd5b



@106 2b95a



@107 2b563



@108 2b175



@109 2ad91



@10a 2a9b6



@10b 2a5e4



@10c 2a21c



@10d 29e5c



@10e 29aa5



@10f 296f7



@110 29352



@111 28fb5



@112 28c21



@113 28894



@114 28510



@115 28195



@116 27e21



@117 27ab5



@118 27751



@119 273f4



@11a 2709f



@11b 26d52



@11c 26a0c



@11d 266cd



@11e 26396



@11f 26066



@120 25d3d



@121 25a1b



@122 25700



@123 253eb



@124 250de



@125 24dd7



@126 24ad6



@127 247dc



@128 244e9



@129 241fc



@12a 23f15



@12b 23c34



@12c 23959



@12d 23685



@12e 233b6



@12f 230ee



@130 22e2b



@131 22b6e



@132 228b6



@133 22605



@134 22359



@135 220b2



@136 21e11



@137 21b75



@138 218df



@139 2164e



@13a 213c2



@13b 2113b



@13c 20eba



@13d 20c3d



@13e 209c6



@13f 20753



@140 204e5



@141 2027c



@142 20018



@143 1fdb9



@144 1fb5e



@145 1f908



@146 1f6b7



@147 1f46a



@148 1f221



@149 1efdd



@14a 1ed9d



@14b 1eb62



@14c 1e92b



@14d 1e6f8



@14e 1e4c9



@14f 1e29f



@150 1e079



@151 1de56



@152 1dc38



@153 1da1e



@154 1d808



@155 1d5f5



@156 1d3e7



@157 1d1dc



@158 1cfd5



@159 1cdd2



@15a 1cbd3



@15b 1c9d7



@15c 1c7df



@15d 1c5ea



@15e 1c3f9



@15f 1c20c



@160 1c022



@161 1be3b



@162 1bc59



@163 1ba79



@164 1b89c



@165 1b6c4



@166 1b4ee



@167 1b31c



@168 1b14c



@169 1af80



@16a 1adb8



@16b 1abf2



@16c 1aa30



@16d 1a870



@16e 1a6b4



@16f 1a4fa



@170 1a344



@171 1a191



@172 19fe0



@173 19e32



@174 19c88



@175 19ae0



@176 1993b



@177 19799



@178 195f9



@179 1945d



@17a 192c2



@17b 1912b



@17c 18f97



@17d 18e05



@17e 18c75



@17f 18ae9



@180 1895f



@181 187d7



@182 18652



@183 184cf



@184 1834f



@185 181d2



@186 18057



@187 17ede



@188 17d68



@189 17bf4



@18a 17a82



@18b 17913



@18c 177a6



@18d 1763b



@18e 174d3



@18f 1736c



@190 17209



@191 170a7



@192 16f47



@193 16dea



@194 16c8f



@195 16b36



@196 169df



@197 1688a



@198 16737



@199 165e7



@19a 16498



@19b 1634c



@19c 16201



@19d 160b9



@19e 15f72



@19f 15e2e



@1a0 15ceb



@1a1 15baa



@1a2 15a6b



@1a3 1592e



@1a4 157f3



@1a5 156ba



@1a6 15583



@1a7 1544d



@1a8 1531a



@1a9 151e8



@1aa 150b8



@1ab 14f8a



@1ac 14e5d



@1ad 14d32



@1ae 14c09



@1af 14ae2



@1b0 149bc



@1b1 14898



@1b2 14776



@1b3 14656



@1b4 14537



@1b5 14419



@1b6 142fe



@1b7 141e4



@1b8 140cb



@1b9 13fb4



@1ba 13e9f



@1bb 13d8b



@1bc 13c79



@1bd 13b69



@1be 13a5a



@1bf 1394c



@1c0 13840



@1c1 13735



@1c2 1362c



@1c3 13524



@1c4 1341e



@1c5 13319



@1c6 13216



@1c7 13114



@1c8 13014



@1c9 12f15



@1ca 12e17



@1cb 12d1b



@1cc 12c20



@1cd 12b26



@1ce 12a2e



@1cf 12937



@1d0 12841



@1d1 1274d



@1d2 1265a



@1d3 12569



@1d4 12478



@1d5 12389



@1d6 1229b



@1d7 121af



@1d8 120c4



@1d9 11fd9



@1da 11ef0



@1db 11e09



@1dc 11d22



@1dd 11c3d



@1de 11b59



@1df 11a77



@1e0 11995



@1e1 118b5



@1e2 117d5



@1e3 116f7



@1e4 1161a



@1e5 1153e



@1e6 11463



@1e7 1138a



@1e8 112b1



@1e9 111da



@1ea 11104



@1eb 1102f



@1ec 10f5b



@1ed 10e87



@1ee 10db6



@1ef 10ce5



@1f0 10c15



@1f1 10b46



@1f2 10a78



@1f3 109ab



@1f4 108e0



@1f5 10815



@1f6 1074b



@1f7 10683



@1f8 105bb



@1f9 104f4



@1fa 1042e



@1fb 1036a



@1fc 102a6



@1fd 101e3



@1fe 10121



@1ff 10060







C—COEFFICIENTS FOR ODD SQUARE ROOT RECIPROCAL









// c0 SQRT odd



width = 28



@000 ffffe7e



@001 ff8047d



@002 ff0165d



@003 fe83403



@004 fe05d50



@005 fd89229



@006 fd0d271



@007 fc91e0c



@008 fc174de



@009 fb9d6cc



@00a fb243be



@00b faabb96



@00c fa33e3b



@00d f9bcb94



@00e f946386



@00f f8d05fb



@010 f85b2d8



@011 f7e6a04



@012 f772b69



@013 f6ff6ed



@014 f68cc7a



@015 f61abf9



@016 f5a9550



@017 f53886d



@018 f4c8538



@019 f458b99



@01a f3e9b7d



@01b f37b4cb



@01c f30d770



@01d f2a0359



@01e f23386e



@01f f1c769b



@020 f15bdce



@021 f0f0def



@022 f0866f0



@023 f01c8b8



@024 efb3335



@025 ef4a658



@026 eee2208



@027 ee7a637



@028 ee132d2



@029 edac7c3



@02a ed464fe



@02b ece0a6c



@02c ec7b7fe



@02d ec16da4



@02e ebb2b4b



@02f eb4f0e1



@030 eaebe57



@031 ea8939b



@032 ea270a1



@033 e9c5553



@034 e9641a6



@035 e903585



@036 e8a30e5



@037 e8433b4



@038 e7e3de4



@039 e784f65



@03a e726829



@03b e6c8820



@03c e66af3d



@03d e60dd70



@03e e5b12ac



@03f e554ee2



@040 e4f9204



@041 e49dc05



@042 e442cd5



@043 e3e846b



@044 e38e2b6



@045 e3347aa



@046 e2db339



@047 e282556



@048 e229df7



@049 e1d1d0c



@04a e17a289



@04b e122e64



@04c e0cc08e



@04d e0758fc



@04e e01f7a1



@04f dfc9c73



@050 df74765



@051 df1f86c



@052 decaf7b



@053 de76c8a



@054 de22f89



@055 ddcf871



@056 dd7c733



@057 dd29bc7



@058 dcd7621



@059 dc85639



@05a dc33c00



@05b dbe276e



@05c db91879



@05d db40f14



@05e daf0b37



@05f daa0cd9



@060 da513ed



@061 da0206c



@062 d9b324a



@063 d96497f



@064 d9165ff



@065 d8c87c3



@066 d87aec2



@067 d82daf0



@068 d7e0c46



@069 d7942b8



@06a d747e40



@06b d6fbed4



@06c d6b046b



@06d d664efc



@06e d619e81



@06f d5cf2ec



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C—COEFFICENTS FOR EVEN SQUARE ROOT RECIPROCAL









// c0 SQRT



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## Claims

1. A system for evaluating a rounded arithmetic expression comprising:
a plurality of tables populated with values to generate a piecewise monotonic function;
an arithmetic unit comprising non-iterative logic coupled to the plurality of tables, the arithmetic unit comprising an input and an output, the input to receive an operand and the output to provide a floating point result for the arithmetic expression comprising an accuracy to a Unit in the Last Place (ULP);
a register comprising an input coupled to the output of the arithmetic unit.

2. The system of claim 1, wherein the arithmetic expression is a reciprocal.

3. The system of claim 1, wherein the arithmetic expression is a square root reciprocal.

4. The system of claim 1, wherein the arithmetic expression is a square root.

5. An arithmetic processor comprising:
a lookup table system including first, second and third component tables configured to provide a first operand, a second operand, a third operand, and a square operand;
a first multiplier comprising an input to receive at least a first portion of an input operand, the first multiplier further coupled to the first component table to multiply the first operand and a square operand to provide a first result, the square operand determined responsively to the first portion of the input operand;
a second multiplier comprising an input to receive at least a second portion of the input operand, the second multiplier further coupled to the second component table to multiply the second operand and a multiplier operand to provide a second result, the multiplier operand determined responsively to the second portion of the input operand; and
an adding circuit configured to add the first result and the second result and the third operand, the third operand determined responsive to a third portion of the input operand to provide a third result;
a rounding circuit coupled to receive the third result and to provide a rounded result accurate to a unit in the last place.

6. The arithmetic processor, as recited in claim 5, further comprising:
a square table configured to provide the square operand.

7. The arithmetic processor, as recited in claim 6, wherein the entries in the square table are stored in a Booth recoded format.

8. The arithmetic processor, as recited in claim 5, further comprising:
a Booth recoder configured to provide the multiplier operand.

9. The arithmetic processor, as recited in claim 5, wherein the arithmetic processor is configured to provide a reciprocal value of an input operand.

10. The arithmetic processor, as recited in claim 5, wherein the arithmetic processor is configured to provide a square root value of an input operand.

11. The arithmetic processor as recited in claim 5, wherein the arithmetic processor is configured to provide a square root reciprocal value of an input operand.

12. The arithmetic processor, as recited in claim 5, wherein the first portion of the input operand comprises high order bits of the input operand.

13. The arithmetic processor, as recited in claim 12, wherein the first and second portions of the input operand are mutually exclusive with each other.

14. The arithmetic processor, as recited in claim 13, wherein the first and third portions of the input operand are mutually exclusive with each other.

15. The arithmetic processor, as recited in claim 14, wherein the second and third portions of the input operand overlap with each other.