# Patent

## Title
Increased precision in the computation of a reciprocal square root

## Abstract

A method and arrangements for increased precision in the computation of a reciprocal square root is disclosed. In accordance with the present invention, it is possible to achieve fifty three (53) bits of precision in less processing time than previously possible.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application is a continuation application of U.S. patent application Ser. No. 10/632,362 filed on Jul. 31, 2003 now U.S. Pat. No. 7,313,584, the contents of which are hereby fully incorporated by reference in its entirety.


FIELD OF THE INVENTION
The present invention relates to increased precision for the computation of a reciprocal square root.
BACKGROUND OF THE INVENTION
In microprocessor design, it is not unusual for the designer of the chip to specify that certain functions are to be performed by the chip. The implementation of the specified functions is then left to another designer. Two such functions which are specified for some microprocessors are the square root function ‘sqrt(x)’ and the reciprocal square root function ‘1/sqrt(x)’. One microprocessor family for which these functions have been specified and implemented is the IBM PowerPC. Such a microprocessor is used in the IBM Blue Gene/L Supercomputer (“BG/L”). See [http://]www.ibm.com/chips/products.powerpc/newsletter/aug2001/new-prod3.html.
The reciprocal square root function is necessary in a number of calculations used in a variety of applications, however, it generally is used in connection with determining the direction of the vector between any two points in space. By way of example, such a function is used in calculating the direction and magnitude of the force between pairs of atoms when simulating the motion of protein molecules in water solution. The function is also used in calculating the best estimate of the rotation and shift between a pair of images of a triangle, i.e., where the triangle might be defined by 3 points picked out on a digital image, such as an image of a fingerprint; for the purpose of matching a ‘candidate’ fingerprint in a large set of ‘reference’ fingerprints.
While the reciprocal square root function may be implemented in a number of ways, there is no standard for its precision. The function should optimally return the double-precision floating point number nearest to the reciprocal of the square root of its argument ‘x’. Compare IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754). ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic, IEEE, New York, 1985. To arrive at such a result, however, requires significant computational resources such as processing time.
In most computational situations, however, it is sufficient to generate an approximation of the reciprocal square root of a number that is precise to some number of bits smaller than the standard fifty three (53) bits. Known implementations of the reciprocal square root function involve a trade-off between precision and computational resources, i.e., processing time.
There thus is a need for a method and system for calculating the reciprocal of a square root of a number that provides for both greater accuracy and greater precision without increasing the need for computing time and resources.
SUMMARY OF THE INVENTION
In accordance with at least one presently preferred embodiment of the present invention there is now broadly contemplated increased precision in the computation of the reciprocal square root of a number
One aspect of the present invention provides a method of for calculating the reciprocal square root of a number, comprising the steps of: forming a piecewise-linear estimate for the reciprocal square root of a number; rounding said estimate to a lower precision; computing the residual of said rounded estimate; using a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and multiplying said rounded estimate by said residual error and adding the result to said rounded estimate.
Another aspect of the present invention provides an apparatus for calculating the reciprocal square root of a number, comprising: an arrangement for forming a piecewise-linear estimate for the reciprocal square root of a number; an arrangement for rounding said estimate to a lower precision; an arrangement for computing the residual of said rounded estimate; an arrangement for using a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and an arrangement for multiplying said rounded estimate by said residual error and adding the result to said rounded estimate.
Furthermore, an additional aspect of the present invention provides A program storage device readable by machine, tangibly embodying a program of instructions executable by the machine to perform a method for calculating the reciprocal square root of a number, comprising the steps of: forming a piecewise-linear estimate for the reciprocal square root of a number; rounding said estimate to a lower precision; computing the residual of said rounded estimate; using a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and multiplying said rounded estimate by said residual error and adding the result to said rounded estimate.
For a better understanding of the present invention, together with other and further features and advantages thereof, reference is made to the following description, taken in conjunction with the accompanying drawings, and the scope of the invention will be pointed out in the appended claims.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flow diagram of the PowerPC implementation of the process for determining the reciprocal square root of the argument ‘X’.
FIG. 2 is a graph diagram of the values returned for the piecewise-linear estimate for the reciprocal square root of a number in the range of 1 to 2 and 2 to 4.
FIG. 3 is a flow diagram of a process involving the determination of the reciprocal square root in conformity with the present invention.
FIG. 4 is a more particular flow diagram of a process involving the determination of the reciprocal square root of 9 in conformity with the present invention.
FIG. 5 depicts a microprocessor suitable for implementing the process of determining the reciprocal square root in conformity with the present invention.



DESCRIPTION OF THE PREFERRED EMBODIMENTS
As previously discussed, IBM PowerPC processors all contain a ‘reciprocal square root estimate’. Referring now to FIG. 1, a piecewise-linear estimate for the reciprocal square root is formed initially. In this implementation of the function, at S100, the argument is first normalized (multiplied by a power of 4) into a range of 1<=x<4. Next, at S110, the top five bits (after the implied leading ‘1’) of the mantissa are used to index one of two pairs of 32-element tables, depending on whether x is in the range ‘1<=x2’ or in the range ‘2<=x4’. This results in slope and offset values ‘m’ and ‘c’, respectively, appropriate for range ‘x’. At S120, The value ‘m*x+c’ is calculated and, at S130, the exponent is adjusted for the initial normalization. At S140, to get from this estimate to the desired result one of two well-known conventional methods is generally used—the Newton-Raphson Iteration or the Taylor Series Expansion.
The process of forming a piecewise-linear estimate is described in S100-S130, is discussed below, and is well known in the art. See Abromowitz and Stegun, Handbook of Mathematical Functions, (1964). FIG. 2 illustrates the graph diagram for the piecewise-linear estimate for the reciprocal square root of a number in the range of 1 to 2 and 2 to 4. As can be seen, the process of forming the estimate involves splitting the region from 1 to 2 into 2(two) sections and the region from 2 to 4 into 2 (two) sections. The process of rounding causes the graph lines to become staircase progressions instead of the straight lines depicted in FIG. 2. As discussed above, once the piecewise-linear estimate is formed, the estimate is usually adjusted by applying Newton's Method or performing a Taylor Expansion.
The Newton-Raphson iteration (also called “Newton's Method”) is well known and is discussed in detail in Abromowitz and Stegun, Handbook of Mathematical Functions, (1964), p. 18, which is hereby incorporated by reference. Newton's Method recognizes that the reciprocal square root of ‘a’ is the solution of the formula a*x*x−1=0. The solution is derived through a few iterations of the formula. The Taylor Series is also well known and is also described in particularity in Abromowitz (p. 15), which is also hereby incorporated by reference. In the Taylor Series, the estimate ‘x0’ of the reciprocal square root is adjusted for more accuracy using an error term ‘e’ as follows. The equation a*x0*x0−1 is solved and a correction term ‘epr’ is developed solving the equation ‘epr=(1+e)**(−0.5)−1’. In the result, ‘x0+(x0*epr), ‘e’ will be small (less than 2**−13 in the BG/L implementation), so the first four (4) or so terms of the asymptotic polynomial expansion for ‘epr’ will be sufficient to achieve the desired precision.
The PowerPC processor defines a ‘floating point multiply-add’ instruction, which computes ‘a*b+c’ for 53-bit−precise arguments and returns a 53-bit-precise result. Using the ‘floating-point multiply-add instruction’ present in the IBM PowerPC and similar processors, the intermediate arithmetic calculation of ‘a*b’ is carried to 106 bits of precision. This gives extended precision for cases where ‘a*b’ and ‘c’ are nearly equal in magnitude but of opposite sign. In the case of the ‘square root’ function and the ‘reciprocal function’, this instruction can provide good accuracy in approximating the solutions for the equations ‘x*x−a=0’ and ‘a*x−1=0’. The merged multiply-add with a result near 0 is apparent from the formulation, and is exploited to bring the results to full 53-bit precision.
In determining the ‘reciprocal square root’ of a number, the Newton-Raphson method uses two multiplications and an addition. PowerPC rounds the result of this first multiplication to 53 bits of precision, which upsets the precision of the final result. As a consequence, in approximately 30% of the cases, successive Newton-Raphson iterations fail to converge upon the correct result, instead oscillating between a number greater than the correct result and lower then the correct result. Further, when using the Taylor Expansion, this rounding off to 53 bits of precision results in an error term ‘e’ that is insufficient to correct the approximation error, thus in 20% of the cases, the Taylor Expansion fails to provide a desired result.
Referring now to FIG. 3, the process for calculating the reciprocal square root of a number in accordance with the present invention is depicted. As was earlier described in S100 through S130 of FIG. 1, and as further illustrated in FIG. 2, the process depicted in FIG. 3 begins by forming a piecewise-linear estimate. At S300, a piecewise-linear estimate for the reciprocal of the square root of ‘x’ is formed by multiplying x by a power 4 into a range of 1<=x<4. The top 5 bits of the mantissa are used to index one of two pairs of 32-element tables where the pairs are slope ‘m’ and offset ‘c’. It will be appreciated that more or less than the top 5 bits of the mantissa may be used depending on the microprocessor's precision. The values for ‘m’ and ‘c’ are looked up in the appropriate table depending on whether 1<=x<2 or 2<=x<4. Next, in S320, the estimate is rounded/truncated to one half of the microprocessor's precision or less than one half. It will be appreciated that in one preferred embodiment of the invention the rounding/truncating of step S320 may be performed to a least one half of the microprocessor's precision, but, in many cases may be performed to less than one half. In S340, the residual is computed by so that the rounded/truncated estimate is multiplied by itself and the result is then multiplied by the argument ‘x’ and 1.0 is subtracted from the product to obtain the residual error. In S350, the polynomial in the residual error is computed by using a Taylor Expansion where the argument value is the residual error calculated in S340. In S360 the original rounded estimate of S320 is compensated by adding the extended precision intermediate product (residual error) of S350 to the original estimate of S320. In 99.9994% of the time, the result is the IEEE-representable (53-bit) number nearest the infinite precision value for the reciprocal square root of ‘x’. In the other 0.0006% of the time, the result is the IEEE-representable (53-bit) number nearest the infinite precision value for the reciprocal square root of ‘x’ but incorrectly rounded in the least significant bit.
Moving on to FIG. 4, the process for estimating the reciprocal square root of 9 is depicted in accordance with the present invention, assuming a base-10 number system. It should be appreciated that the invention is applicable to any number of bases including binary and hexadecimal numbers. First, at S400, a piecewise-linear estimate for the reciprocal square root of 9 is obtained by finding the values for A and B using the equation A+B*9. In the example, the value is 0.3234. Next, at S410, this value is then rounded to two decimal places to obtain a new estimate of 0.32. At S420, the calculation is as follows: 0.3200×0.3200=0.1024, 0.1024×9.000−1.000=−0.07840. At S430, a Taylor Expansion is performed and the polynomial in the residual of −0.07840 is calculated to the desired number of terms as follows, using the polynomial equation f(x)=x*(−1⁄2+x*(− 5/16+x* 35/128))) where x=−0.07840, f(−0.07840)=0.04167. At S440, the result of the Taylor Expansion is used to compensate the original rounded piecewise-linear estimation as follows: 0.3200*0.04167+0.3200=0.3333.
As can be seen from the above discussion, it is apparent that by rounding off the estimate to half the processor's floating point precision or less than half that precision, the ‘multiply’ operation used to square the rounded estimate is exact in that all the bits that would nominally be dropped when the machine rounds the result are zeroes. This results in a more accurate error factor ‘e’ and provides a more accurate end result.
Thus, in 99.9994% of test cases, the present invention results in a desired result. In the remaining 0.0006%, there is a rounding error in the last significant bit. It will be appreciated that the invention results in a significant improvement over the 70% accuracy provided by the Newton-Raphson Method and the 80% accuracy of the Taylor Expansion without rounding.
Finally, FIG. 5 depicts a microprocessor suitable for implementing the process of determining the reciprocal square root in conformity with the present invention. At 500, the microprocessor is depicted. At 510, the processor function for calculating the reciprocal square root of a number in conformity with the present invention is depicted. In one preferred embodiment of the invention, the microprocessor will be capable of performing calculations with up to 106 bits of precision. However, it will be appreciated that the invention herein is applicable to microprocessors having more or less than the 106 bits of precision assumed herein.
Set forth in the Appendix hereto is a compiler listing, which includes source code written in the C computer language that a programmer would use to instruct a microprocessor or computer to evaluate the reciprocal square root of a number, a timing section timing section which shows how many clock cycles the compiler estimate the program will take, and the sequence of machine instructions to implement the code. The material in the Appendix illustrates how the present invention may be utilized.
It is to be understood that the present invention, in accordance with at least one preferred embodiment, includes an arrangement for forming a piecewise-linear estimate for the reciprocal square root of a number; an arrangement for rounding said estimate to a lower precision; an arrangement for computing the residual of said rounded estimate; an arrangement for using a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and an arrangement for multiplying said rounded estimate by said residual error and adding the result to said rounded estimate. Together these elements may be implemented on at least one general-purpose computer running suitable software programs. These may be implemented on at least one Integrated Circuit or part of at least one Integrated Circuit. Thus, it is to be understood that the invention may be implemented on hardware, software, or a combination of both.
If not otherwise stated herein, it is to be assumed that all patents, patent applications, patent publications and other publications (including web-based publications) mentioned and cited herein are hereby fully incorporated by reference herein as if set forth in their entirety herein.
Although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings, it is to be understood that the invention is not limited to those precise embodiments, and that various other changes and modifications may be affected therein by one skilled in the art without departing from the scope or spirit of the invention.







APPENDIX







VisualAge C++ for Linux on pSeries, Version 6.0.0.0 --- tenrootc.c


07/30/2003 11:41:05 AM (C)


>>>>> SOURCE SECTION <<<<<


     1 | #include <math.h>


     2 | double reciprocal_square_root(double x)


     3 | {


     4 |  return 1.0/sqrt(x) ;


     5 | }


     6 |


     7 | void ten_reciprocal_square_root(double* f, const double* x)


     8 | {


     9 |  double x0 = x[0] ;


    10 |  double x1 = x[1] ;


    11 |  double x2 = x[2] ;


    12 |  double x3 = x[3] ;


    13 |  double x4 = x[4] ;


    14 |  double x5 = x[5] ;


    15 |  double x6 = x[6] ;


    16 |  double x7 = x[7] ;


    17 |  double x8 = x[8] ;


    18 |  double x9 = x[9] ;


    19 |  double r0 = 1.0/sqrt(x0) ;


    20 |  double r1 = 1.0/sqrt(x1) ;


    21 |  double r2 = 1.0/sqrt(x2) ;


    22 |  double r3 = 1.0/sqrt(x3) ;


    23 |  double r4 = 1.0/sqrt(x4) ;


    24 |  double r5 = 1.0/sqrt(x5) ;


    25 |  double r6 = 1.0/sqrt(x6) ;


    26 |  double r7 = 1.0/sqrt(x7) ;


    27 |  double r8 = 1.0/sqrt(x8) ;


    28 |  double r9 = 1.0/sqrt(x9) ;


    29 |  f[0] = r0 ;


    30 |  f[1] = r1 ;


    31 |  f[2] = r2 ;


    32 |  f[3] = r3 ;


    33 |  f[4] = r4 ;


    34 |  f[5] = r5 ;


    35 |  f[6] = r6 ;


    36 |  f[7] = r7 ;


    37 |  f[8] = r8 ;


    38 |  f[9] = r9 ;


    39 | }


    40 |


    41 |


** Procedure List for Proc # 1: ten_reciprocal_square_root End of


Phase 3 **








  0:   HDR



  4:   BB_BEGIN
 2 /  0


  0:   PROC
f,x,gr3,gr4


  0:   DIRCTIV
issue_cycle,0


  0:   LR
gr12=gr1


  0:   LI
gr0=−16


  0:   DIRCTIV
issue_cycle,1


  0:   ST4U
gr1,#stack(gr1,−80)=gr1


  0:   DIRCTIV
issue_cycle,2


  0:   SFPLU
gr12,#stack(gr12,gr0,0)=fp31,fp63


  0:   DIRCTIV
issue_cycle,3


  0:   SFPLU
gr12,#stack(gr12,gr0,0)=fp30,fp62


  0:   DIRCTIV
issue_cycle,4


  0:   SFPLU
gr12,#stack(gr12,gr0,0)=fp29,fp61


  0:   DIRCTIV
issue_cycle,5


  0:   SFPLU
gr12,#stack(gr12,gr0,0)=fp28,fp60


  0:   FENCE


  0:   DIRCTIV
end_prologue


  0:   FENCE


  0:   DIRCTIV
issue_cycle,0


 39:   DIRCTIV
start_epilogue


 18:   LI
gr6=72


 17:   LFL
fp13=(*)Cdouble(gr4,64)


  0:   DIRCTIV
issue_cycle,1


 16:   LI
gr7=56


 18:   LFL
fp45=(*)Cdouble(gr4,gr6,0,trap=72)


  0:   DIRCTIV
issue_cycle,2


 14:   LI
gr5=40


 15:   LFL
fp3=(*)Cdouble(gr4,48)


  0:   DIRCTIV
issue_cycle,3


 16:   LFL
fp35=(*)Cdouble(gr4,gr7,0,trap=56)


 12:   LI
gr6=24


  0:   DIRCTIV
issue_cycle,4


 19:   LA
gr8=.+CONSTANT_AREA%HI(gr2,0)


 13:   LFL
fp1=(*)Cdouble(gr4,32)


  0:   DIRCTIV
issue_cycle,5


 14:   LFL
fp33=(*)Cdouble(gr4,gr5,0,trap=40)


 27:   FPRSQRE
fp12,fp44=fp13,fp45


  0:   DIRCTIV
issue_cycle,6


 11:   LFL
fp31=(*)Cdouble(gr4,16)


 10:   LI
gr7=8


  0:   DIRCTIV
issue_cycle,7


 25:   FPRSQRE
fp11,fp43=fp3,fp35


 12:   LFL
fp63=(*)Cdouble(gr4,gr6,0,trap=24)


  0:   DIRCTIV
issue_cycle,8


 19:   LA
gr9=+CONSTANT_AREA%LO(gr8,0)


  9:   LFL
fp10=(*)Cdouble(gr4,0)


  0:   DIRCTIV
issue_cycle,9


 23:   FPRSQRE
fp9,fp41=fp1,fp33


 10:   LFL
fp42=(*)Cdouble(gr4,gr7,0,trap=8)


  0:   DIRCTIV
issue_cycle,10


 27:   FPMUL
fp4,fp36=fp12,fp44,fp12,fp44,fcr


 19:   LFPS
fp8,fp40=+CONSTANT_AREA(gr9,gr6,0,trap=24)


  0:   DIRCTIV
issue_cycle,11


 19:   LI
gr8=32


 21:   FPRSQRE
fp7,fp39=fp31,fp63


  0:   DIRCTIV
issue_cycle,12


 25:   FPMUL
fp2,fp34=fp11,fp43,fp11,fp43,fcr


 19:   LFS
fp30=+CONSTANT_AREA(gr9,4)


  0:   DIRCTIV
issue_cycle,13


 19:   FPRSQRE
fp6,fp38=fp10,fp42


 19:   LFPS
fp29,fp61=+CONSTANT_AREA(gr9,gr8,0,trap=32)


  0:   DIRCTIV
issue_cycle,14


 23:   FPMUL
fp0,fp32=fp9,fp41,fp9,fp41,fcr


 19:   LFPS
fp28,fp60=+CONSTANT_AREA(gr9,gr5,0,trap=40)


  0:   DIRCTIV
issue_cycle,15


 19:   LI
gr4=48


 27:   FPMADD
fp4,fp36=fp8,fp40,fp13,fp45,fp4,fp36,fcr


  0:   DIRCTIV
issue_cycle,16


 19:   LFPS
fp5,fp37=+CONSTANT_AREA(gr9,gr4,0,trap=48)


 21:   FPMUL
fp13,fp45=fp7,fp39,fp7,fp39,fcr


  0:   DIRCTIV
issue_cycle,17


 25:   FPMADD
fp3,fp35=fp8,fp40,fp3,fp35,fp2,fp34,fcr


 38:   LI
gr6=72


  0:   DIRCTIV
issue_cycle,18


 19:   FPMUL
fp2,fp34=fp6,fp38,fp6,fp38,fcr


 39:   LI
gr0=16


  0:   DIRCTIV
issue_cycle,19


 23:   FPMADD
fp1,fp33=fp8,fp40,fp1,fp33,fp0,fp32,fcr


 39:   LR
gr12=gr1


  0:   DIRCTIV
issue_cycle,20


 27:   FXPMADD
fp0,fp32=fp29,fp61,fp4,fp36,fp30,fp30,fcr


 36:   LI
gr7=56


  0:   DIRCTIV
issue_cycle,21


 21:   FPMADD
fp31,fp63=fp8,fp40,fp31,fp63,fp13,fp45,fcr


  0:   DIRCTIV
issue_cycle,22


 25:   FXPMADD
fp13,fp45=fp29,fp61,fp3,fp35,fp30,fp30,fcr


  0:   DIRCTIV
issue_cycle,23


 19:   FPMADD
fp8,fp40=fp8,fp40,fp10,fp42,fp2,fp34,fcr


  0:   DIRCTIV
issue_cycle,24


 23:   FXPMADD
fp2,fp34=fp29,fp61,fp1,fp33,fp30,fp30,fcr


  0:   DIRCTIV
issue_cycle,25


 27:   FPMADD
fp10,fp42=fp28,fp60,fp4,fp36,fp0,fp32,fcr


  0:   DIRCTIV
issue_cycle,26


 21:   FXPMADD
fp0,fp32=fp29,fp61,fp31,fp63,fp30,fp30,fcr


  0:   DIRCTIV
issue_cycle,27


 25:   FPMADD
fp13,fp45=fp28,fp60,fp3,fp35,fp13,fp45,fcr


  0:   DIRCTIV
issue_cycle,28


 19:   FXPMADD
fp30,fp62=fp29,fp61,fp8,fp40,fp30,fp30,fcr


  0:   DIRCTIV
issue_cycle,29


 23:   FPMADD
fp2,fp34=fp28,fp60,fp1,fp33,fp2,fp34,fcr


  0:   DIRCTIV
issue_cycle,30


 27:   FPMADD
fp10,fp42=fp5,fp37,fp4,fp36,fp10,fp42,fcr


  0:   DIRCTIV
issue_cycle,31


 21:   FPMADD
fp0,fp32=fp28,fp60,fp31,fp63,fp0,fp32,fcr


  0:   DIRCTIV
issue_cycle,32


 25:   FPMADD
fp13,fp45=fp5,fp37,fp3,fp35,fp13,fp45,fcr


  0:   DIRCTIV
issue_cycle,33


 19:   FPMADD
fp30,fp62=fp28,fp60,fp8,fp40,fp30,fp62,fcr


  0:   DIRCTIV
issue_cycle,34


 23:   FPMADD
fp2,fp34=fp5,fp37,fp1,fp33,fp2,fp34,fcr


  0:   DIRCTIV
issue_cycle,35


 27:   FPMUL
fp4,fp36=fp4,fp36,fp10,fp42,fcr


 39:   LFPLU
fp28,fp60,gr12=#stack(gr12,gr0,0)


  0:   DIRCTIV
issue_cycle,36


 21:   FPMADD
fp0,fp32=fp5,fp37,fp31,fp63,fp0,fp32,fcr


 39:   LFPLU
fp29,fp61,gr12=#stack(gr12,gr0,0)


  0:   DIRCTIV
issue_cycle,37


 25:   FPMUL
fp3,fp35=fp3,fp35,fp13,fp45,fcr


  0:   DIRCTIV
issue_cycle,38


 19:   FPMADD
fp5,fp37=fp5,fp37,fp8,fp40,fp30,fp62,fcr


  0:   DIRCTIV
issue_cycle,39


 23:   FPMUL
fp1,fp33=fp1,fp33,fp2,fp34,fcr


  0:   DIRCTIV
issue_cycle,40


 27:   FPMADD
fp2,fp34=fp12,fp44,fp12,fp44,fp4,fp36,fcr


 39:   LFPLU
fp30,fp62,gr12=#stack(gr12,gr0,0)


  0:   DIRCTIV
issue_cycle,41


 21:   FPMUL
fp0,fp32=fp31,fp63,fp0,fp32,fcr


  0:   DIRCTIV
issue_cycle,42


 25:   FPMADD
fp3,fp35=fp11,fp43,fp11,fp43,fp3,fp35,fcr


  0:   DIRCTIV
issue_cycle,43


 19:   FPMUL
fp4,fp36=fp8,fp40,fp5,fp37,fcr


 39:   LFPLU
fp31,fp63,gr12=#stack(gr12,gr0,0)


  0:   DIRCTIV
issue_cycle,44


 23:   FPMADD
fp1,fp33=fp9,fp41,fp9,fp41,fp1,fp33,fcr


 39:   AI
gr1=gr1,80,gr12


  0:   DIRCTIV
issue_cycle,45


 39:   CONSUME
gr1,gr2,lr,gr14-gr31,fp14-fp31,fp46-







fp63,cr[234],fsr,fcr,ctr








 38:   STFL
(*)double(gr3,gr6,0,trap=72)=fp34


 32:   LI
gr6=24


  0:   DIRCTIV
issue_cycle,46


 21:   FPMADD
fp0,fp32=fp7,fp39,fp7,fp39,fp0,fp32,fcr


 37:   STFL
(*)double(gr3,64)=fp2


  0:   DIRCTIV
issue_cycle,47


 36:   STFL
(*)double(gr3,gr7,0,trap=56)=fp35


 30:   LI
gr7=8


  0:   DIRCTIV
issue_cycle,48


 35:   STFL
(*)double(gr3,48)=fp3


  0:   DIRCTIV
issue_cycle,49


 19:   FPMADD
fp2,fp34=fp6,fp38,fp6,fp38,fp4,fp36,fcr


 34:   STFL
(*)double(gr3,gr5,0,trap=40)=fp33


  0:   DIRCTIV
issue_cycle,50


 33:   STFL
(*)double(gr3,32)=fp1


  0:   DIRCTIV
issue_cycle,51


 32:   STFL
(*)double(gr3,gr6,0,trap=24)=fp32


  0:   DIRCTIV
issue_cycle,52


 31:   STFL
(*)double(gr3,16)=fp0


  0:   DIRCTIV
issue_cycle,54


 30:   STFL
(*)double(gr3,gr7,0,trap=8)=fp34


  0:   DIRCTIV
issue_cycle,55


 29:   STFL
(*)double(gr3,0)=fp2


 39:   BA lr


  4:   BB_END


  5:   BB_BEGIN
 3 /  0


 39:   PEND


  5:   BB_END







** End of Procedure List for Proc #  1: ten_reciprocal_square_root End


of Phase 3 **


** Procedure List for Proc #  2: reciprocal_square_root End of Phase 3


**








  0:   HDR



  4:   BB_BEGIN
 2 /  0


  0:   PROC
x,fp1


  0:   FENCE


  0:   DIRCTIV
end_prologue


  0:   FENCE


  0:   DIRCTIV
issue_cycle,0


  5:   DIRCTIV
start_epilogue


  4:   FRSQRE
fp0=fp1


  4:   LA
gr3=.+CONSTANT_AREA%HI(gr2,0)


  0:   DIRCTIV
issue_cycle,1


  4:   LA
gr3=+CONSTANT_AREA%LO(gr3,0)


  0:   DIRCTIV
issue_cycle,2


  4:   LFS
fp2=+CONSTANT_AREA(gr3,0)


  0:   DIRCTIV
issue_cycle,3


  4:   LFS
fp4=+CONSTANT_AREA(gr3,4)


  0:   DIRCTIV
issue_cycle,4


  4:   LFS
fp3=+CONSTANT_AREA(gr3,8)


  0:   DIRCTIV
issue_cycle,5


  4:   MFL
fp5=fp0,fp0,fcr


  4:   LFS
fp6=+CONSTANT_AREA(gr3,12)


  0:   DIRCTIV
issue_cycle,6


  4:   LFS
fp7=+CONSTANT_AREA(gr3,16)


  0:   DIRCTIV
issue_cycle,10


  4:   FMA
fp1=fp2,fp1,fp5,fcr


  0:   DIRCTIV
issue_cycle,15


  4:   FMA
fp2=fp3,fp1,fp4,fcr


  0:   DIRCTIV
issue_cycle,20


  4:   FMA
fp2=fp6,fp1,fp2,fcr


  0:   DIRCTIV
issue_cycle,25


  4:   FMA
fp2=fp7,fp1,fp2,fcr


  0:   DIRCTIV
issue_cycle,30


  4:   MFL
fp1=fp1,fp2,fcr


  0:   DIRCTIV
issue_cycle,35


  4:   FMA
fp1=fp0,fp0,fp1,fcr


  0:   DIRCTIV
issue_cycle,36


  5:   CONSUME
gr1,gr2,lr,gr14-gr31,fp1,fp14-fp31,fp46-







fp63,cr[234],fsr,fcr,ctr








  5:   BA
lr


  4:   BB_END


  5:   BB_BEGIN
 3 /  0


  5:   PEND


  5:   BB_END







** End of Procedure List for Proc #  2: reciprocal_square_root End of


Phase 3 **








GPR's set/used:
ssuu ssss ss-- s--- ---- ---- ---- ----


FPR's set/used:
ssss ssss ssss ss-- ---- ---- ---- ssss



ssss ssss ssss ss-- ---- ---- ---- ssss


CCR's set/used:
---- ----








  | 000000
PDEF







ten_reciprocal_square_root











  0|


PROC
f,x,gr3,gr4


  0| 000000 ori
602C0000
1
LR
gr12=gr1


  0| 000004 addi
3800FFF0
1
LI
gr0=−16


  0| 000008 stwu
9421FFB0
1
ST4U
gr1,#stack(gr1,−80)=gr1


  0| 00000C stfpdux
7FEC07DC
1
SFPLU







gr12,#stack(gr12,gr0,0)=fp31,fp63










  0| 000010 stfpdux
7FCC07DC
1
SFPLU







gr12,#stack(gr12,gr0,0)=fp30,fp62










  0| 000014 stfpdux
7FAC07DC
1
SFPLU







gr12,#stack(gr12,gr0,0)=fp29,fp61










  0| 000018 stfpdux
7F8C07DC
1
SFPLU







gr12,#stack(gr12,gr0,0)=fp28,fp60











 18| 00001C addi
38C00048
1
LI
gr6=72


 17| 000020 lfd
C9A40040
1
LFL
fp13=(*)Cdouble(gr4,64)


 16| 000024 addi
38E00038
1
LI
gr7=56


 18| 000028 lfsdx
7DA4319C
1
LFL







fp45=(*)Cdouble(gr4,gr6,0,trap=72)











 14| 00002C addi
38A00028
1
LI
gr5=40


 15| 000030 lfd
C8640030
1
LFL
fp3=(*)Cdouble(gr4,48)


 16| 000034 lfsdx
7C64399C
1
LFL







fp35=(*)Cdouble(gr4,gr7,0,trap=56)











 12| 000038 addi
38C00018
1
LI
gr6=24


 19| 00003C addis
3D000000
1
LA







gr8=.+CONSTANT_AREA%HI(gr2,0)











 13| 000040 lfd
C8240020
1
LFL
fp1=(*)Cdouble(gr4,32)


 14| 000044 lfsdx
7C24299C
1
LFL







fp33=(*)Cdouble(gr4,gr5,0,trap=40)











 27| 000048 fprsqrte
0180681E
1
FPRSQRE
fp12,fp44=fp13,fp45


 11| 00004C lfd
CBE40010
1
LFL
fp31=(*)Cdouble(gr4,16)


 10| 000050 addi
38E00008
1
LI
gr7=8


 25| 000054 fprsqrte
0160181E
1
FPRSQRE
fp11,fp43=fp3,fp35


 12| 000058 lfsdx
7FE4319C
1
LFL







fp63=(*)Cdouble(gr4,gr6,0,trap=24)










 19| 00005C addi
39280000
1
LA







gr9=+CONSTANT_AREA%LO(gr8,0)











  9| 000060 lfd
C9440000
1
LFL
fp10=(*)Cdouble(gr4,0)


 23| 000064 fprsqrte
0120081E
1
FPRSQRE
fp9,fp41=fp1,fp33


 10| 000068 lfsdx
7D44399C
1
LFL







fp42=(*)Cdouble(gr4,gr7,0,trap=8)










 27| 00006C fpmul
008C0310
1
FPMUL







fp4,fp36=fp12,fp44,fp12,fp44,fcr










 19| 000070 lfpsx
7D09331C
1
LFPS







fp8,fp40=+CONSTANT_AREA(gr9,gr6,0,trap=24)











 19| 000074 addi
39000020
1
LI
gr8=32


 21| 000078 fprsqrte
00E0F81E
1
FPRSQRE
fp7,fp39=fp31,fp63


 25| 00007C fpmul
004B02D0
1
FPMUL







fp2,fp34=fp11,fp43,fp11,fp43,fcr










 19| 000080 lfs
C3C90004
1
LFS







fp30=+CONSTANT_AREA(gr9,4)











 19| 000084 fprsqrte
00C0501E
1
FPRSQRE
fp6,fp38=fp10,fp42


 19| 000088 lfpsx
7FA9431C
1
LFPS







fp29,fp61=+CONSTANT_AREA(gr9,gr8,0,trap=32)










 23| 00008C fpmul
00090250
1
FPMUL







fp0,fp32=fp9,fp41,fp9,fp41,fcr










 19| 000090 lfpsx
7F892B1C
1
LFPS







fp28,fp60=+CONSTANT_AREA(gr9,gr5,0,trap=40)











 19| 000094 addi
38800030
1
LI
gr4=48


 27| 000098 fpmadd
008D4120
1
FPMADD







fp4,fp36=fp8,fp40,fp13,fp45,fp4,fp36,fcr










 19| 00009C lfpsx
7CA9231C
1
LFPS







fp5,fp37=+CONSTANT_AREA(gr9,gr4,0,trap=48)










 21| 0000A0 fpmul
01A701D0
1
FPMUL







fp13,fp45=fp7,fp39,fp7,fp39,fcr










 25| 0000A4 fpmadd
006340A0
1
FPMADD







fp3,fp35=fp8,fp40,fp3,fp35,fp2,fp34,fcr











 38| 0000A8 addi
38C00048
1
LI
gr6=72


 19| 0000AC fpmul
00460190
1
FPMUL







fp2,fp34=fp6,fp38,fp6,fp38,fcr











 39| 0000B0 addi
38000010
1
LI
gr0=16


 23| 0000B4 fpmadd
00214020
1
FPMADD







fp1,fp33=fp8,fp40,fp1,fp33,fp0,fp32,fcr











 39| 0000B8 ori
602C0000
1
LR
gr12=gr1


 27| 0000BC fxcpmadd
001EE924
1
FXPMADD







fp0,fp32=fp29,fp61,fp4,fp36,fp30,fp30,fcr











 36| 0000C0 addi
38E00038
1
LI
gr7=56


 21| 0000C4 fpmadd
03FF4360
1
FPMADD







fp31,fp63=fp8,fp40,fp31,fp63,fp13,fp45,fcr










 25| 0000C8 fxcpmadd
01BEE8E4
1
FXPMADD







fp13,fp45=fp29,fp61,fp3,fp35,fp30,fp30,fcr










 19| 0000CC fpmadd
010A40A0
1
FPMADD







fp8,fp40=fp8,fp40,fp10,fp42,fp2,fp34,fcr










 23| 0000D0 fxcpmadd
005EE864
1
FXPMADD







fp2,fp34=fp29,fp61,fp1,fp33,fp30,fp30,fcr










 27| 0000D4 fpmadd
0144E020
1
FPMADD







fp10,fp42=fp28,fp60,fp4,fp36,fp0,fp32,fcr










 21| 0000D8 fxcpmadd
001EEFE4
1
FXPMADD







fp0,fp32=fp29,fp61,fp31,fp63,fp30,fp30,fcr










 25| 0000DC fpmadd
01A3E360
1
FPMADD







fp13,fp45=fp28,fp60,fp3,fp35,fp13,fp45,fcr










 19| 0000E0 fxcpmadd
03DEEA24
1
FXPMADD







fp30,fp62=fp29,fp61,fp8,fp40,fp30,fp30,fcr










 23| 0000E4 fpmadd
0041E0A0
1
FPMADD







fp2,fp34=fp28,fp60,fp1,fp33,fp2,fp34,fcr










 27| 0000E8 fpmadd
01442AA0
1
FPMADD







fp10,fp42=fp5,fp37,fp4,fp36,fp10,fp42,fcr










 21| 0000EC fpmadd
001FE020
1
FPMADD







fp0,fp32=fp28,fp60,fp31,fp63,fp0,fp32,fcr










 25| 0000F0 fpmadd
01A32B60
1
FPMADD







fp13,fp45=fp5,fp37,fp3,fp35,fp13,fp45,fcr










 19| 0000F4 fpmadd
03C8E7A0
1
FPMADD







fp30,fp62=fp28,fp60,fp8,fp40,fp30,fp62,fcr










 23| 0000F8 fpmadd
004128A0
1
FPMADD







fp2,fp34=fp5,fp37,fp1,fp33,fp2,fp34,fcr










 27| 0000FC fpmul
00840290
1
FPMUL







fp4,fp36=fp4,fp36,fp10,fp42,fcr










 39| 000100 lfpdux
7F8C03DC
1
LFPLU







fp28,fp60,gr12=#stack(gr12,gr0,0)










 21| 000104 fpmadd
001F2820
1
FPMADD







fp0,fp32=fp5,fp37,fp31,fp63,fp0,fp32,fcr










 39| 000108 lfpdux
7FAC03DC
1
LFPLU







fp29,fp61,gr12=#stack(gr12,gr0,0)










 25| 00010C fpmul
00630350
1
FPMUL







fp3,fp35=fp3,fp35,fp13,fp45,fcr










 19| 000110 fpmadd
00A82FA0
1
FPMADD







fp5,fp37=fp5,fp37,fp8,fp40,fp30,fp62,fcr










 23| 000114 fpmul
00210090
1
FPMUL







fp1,fp33=fp1,fp33,fp2,fp34,fcr










 27| 000118 fpmadd
004C6120
1
FPMADD







fp2,fp34=fp12,fp44,fp12,fp44,fp4,fp36,fcr










 39| 00011C lfpdux
7FCC03DC
1
LFPLU







fp30,fp62,gr12=#stack(gr12,gr0,0)










 21| 000120 fpmul
001F0010
1
FPMUL







fp0,fp32=fp31,fp63,fp0,fp32,fcr










 25| 000124 fpmadd
006B58E0
1
FPMADD







fp3,fp35=fp11,fp43,fp11,fp43,fp3,fp35,fcr










 19| 000128 fpmul
00880150
1
FPMUL







fp4,fp36=fp8,fp40,fp5,fp37,fcr










 39| 00012C lfpdux
7FEC03DC
1
LFPLU







fp31,fp63,gr12=#stack(gr12,gr0,0)










 23| 000130 fpmadd
00294860
1
FPMADD







fp1,fp33=fp9,fp41,fp9,fp41,fp1,fp33,fcr











 39| 000134 addi
38210050
1
AI
gr1=gr1,80,gr12


 38| 000138 stfsdx
7C43359C
1
STFL







(*)double(gr3,gr6,0,trap=72)=fp34











 32| 00013C addi
38C00018
1
LI
gr6=24


 21| 000140 fpmadd
00073820
1
FPMADD







fp0,fp32=fp7,fp39,fp7,fp39,fp0,fp32,fcr











 37| 000144 stfd
D8430040
1
STFL
(*)double(gr3,64)=fp2


 36| 000148 stfsdx
7C633D9C
1
STFL







(*)double(gr3,gr7,0,trap=56)=fp35











 30| 00014C addi
38E00008
1
LI
gr7=8


 35| 000150 stfd
D8630030
1
STFL
(*)double(gr3,48)=fp3


 19| 000154 fpmadd
00463120
1
FPMADD







fp2,fp34=fp6,fp38,fp6,fp38,fp4,fp36,fcr










 34| 000158 stfsdx
7C232D9C
1
STFL







(*)double(gr3,gr5,0,trap=40)=fp33











 33| 00015C stfd
D8230020
1
STFL
(*)double(gr3,32)=fp1


 32| 000160 stfsdx
7C03359C
1
STFL







(*)double(gr3,gr6,0,trap=24)=fp32











 31| 000164 stfd
D8030010
1
STFL
(*)double(gr3,16)=fp0


 30| 000168 stfsdx
7C433D9C
1
STFL







(*)double(gr3,gr7,0,trap=8)=fp34











 29| 00016C stfd
D8430000
1
STFL
(*)double(gr3,0)=fp2


 39| 000170 bclr
4E800020
0
BA
lr








  |         Instruction count
93








GPR's set/used:
--us ---- ---- ---- ---- ---- ---- ----


FPR's set/used:
ssss ssss ---- ---- ---- ---- ---- ----



---- ---- ---- ---- ---- ---- ---- ----


CCR's set/used:
---- ----











  | 000000


PDEF
reciprocal_square_root


  0|


PROC
x,fp1


  4| 000174 frsqrte
FC000834
1
FRSQRE
fp0=fp1


  4| 000178 addis
3C600000
1
LA







gr3=.+CONSTANT_AREA%HI(gr2,0)










  4| 00017C addi
38630000
1
LA







gr3=+CONSTANT_AREA%LO(gr3,0)










  4| 000180 lfs
C0430000
1
LFS







fp2=+CONSTANT_AREA(gr3,0)










  4| 000184 lfs
C0830004
1
LFS







fp4=+CONSTANT_AREA(gr3,4)










  4| 000188 lfs
C0630008
1
LFS







fp3=+CONSTANT_AREA(gr3,8)











  4| 00018C fmul
FCA00032
1
MFL
fp5=fp0,fp0,fcr


  4| 000190 lfs
C0C3000C
1
LFS







fp6=+CONSTANT_AREA(gr3,12)










  4| 000194 lfs
C0E30010
1
LFS







fp7=+CONSTANT_AREA(gr3,16)











  4| 000198 fmadd
FC21117A
2
FMA
fp1=fp2,fp1,fp5,fcr


  4| 00019C fmadd
FC41193A
4
FMA
fp2=fp3,fp1,fp4,fcr


  4| 0001A0 fmadd
FC4130BA
4
FMA
fp2=fp6,fp1,fp2,fcr


  4| 0001A4 fmadd
FC4138BA
4
FMA
fp2=fp7,fp1,fp2,fcr


  4| 0001A8 fmul
FC2100B2
4
MFL
fp1=fp1,fp2,fcr


  4| 0001AC fmadd
FC20007A
4
FMA
fp1=fp0,fp0,fp1,fcr


  5| 0001B0 bclr
4E800020
0
BA lr







  |        Instruction count 16


  |        Constant Area


  | 000000    BF800000 3E8C0000 BEA00000 3EC00000 BF000000


49424D20


  | 000018    BF800000 BF800000 BEA00000 BEA00000 3EC00000


3EC00000


  | 000030    BF000000 BF000000

## Claims

1. A method for calculating the reciprocal square root of a number in a computing device comprising a processor having an embedded program for executing a reciprocal square root estimate operation, the reciprocal square root estimate operation comprising the steps of:
using the processor to form a piecewise-linear estimate for the reciprocal square root of a number;
using the processor to round said estimate to a lower precision;
using the processor to compute the residual of said rounded estimate;
using the processor to utilize a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and
using the processor to multiply said rounded estimate by said residual error and adding the result to said rounded estimate.

2. The method of claim 1, said estimate is rounded to one half the number of digits sought for the final result.

3. The method of claim 1, wherein said estimate is rounded to less than one half of the number of digits sought for the final result.

4. An apparatus for calculating the reciprocal square root of a number, comprising:
one or more processors;
a memory in operative connection with the one or more processors;
wherein, responsive to execution of program instructions accessible to the one or more processors, the one or more processors are configured to:
form a piecewise-linear estimate for the reciprocal square root of a number;
round said estimate to a lower precision;
compute the residual of said rounded estimate;
use a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and
multiply said rounded estimate by said residual error and adding the result to said rounded estimate.

5. The apparatus of claim 4, wherein said estimate is rounded to one half the number of digits sought for the final result.

6. The apparatus of claim 4, wherein the said estimate is rounded to less than one half the number of digits sought for the final result.

7. A non-transitory program storage device readable by machine, tangibly embodying a program of instructions executable by the machine to perform a method for calculating the reciprocal square root of a number, comprising the steps of:
forming a piecewise-linear estimate for the reciprocal square root of a number;
rounding said estimate to a lower precision;
computing the residual of said rounded estimate;
using a Taylor Expansion to compute the polynomial in said residual of said estimate to obtain the residual error; and
multiplying said rounded estimate by said residual error and adding the result to said rounded estimate.

8. The program storage device of claim 7, wherein said estimate is rounded to one half the number of digits sought for the final result.

9. The program storage device of claim 7, wherein said estimate is rounded to less than one half the number of digits sought for the final result.