CA2453712A1 - Method of performing binary integer division - Google Patents
Method of performing binary integer division Download PDFInfo
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- G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
- G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
- G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices
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Abstract
There is provided a method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method including: (a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders; wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d; q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
Description
METHOD OF PERFORMING BINARY INTEGER DIVISION
Field of Invention The present invention relates to the field of mathematical processing in computer systems and more particularly to methods of performing binary integer division.
Back round The execution of binary integer division in many computer systems requires considerably more processing cycles than other operations such as binary integer multiplication. As a result, methods have been developed for computing binary integer division by using low-cost operations, in terms of processor cycles and execution time required, such as shift, multiply and add rather than hardware based integer division. Binary integer division methods are used in compilers for generating program code used to evaluate integer divisions, for example, in programming loops with compile-time known loop-invariant divisors (i.e., the divisor is the same for each division).
Programs resulting from implementation of these methods typically run faster than programs using the native hardware binary integer division, thus improving execution time and overall performance.
However, when the divisor of a binary integer division operation used in a program loop is Ioop-invariant, but its value is not known at compile time, changes to traditional processing methods are required.
Surnmary of Invention In accordance with one aspect of the present invention, there is provided a method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method including: (a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders; wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d; q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
In accordance with another aspect of the present invention, there is provided a method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method including: (a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d; (6) padding the quotient value q(i) to the left with binary 0's equal to N less the number of bits in the intermediate Ct~9-2003-0101 2 remainder r(i); (c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d); (d) padding the subsequent intermediate remainder r(i+1 ) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from tale zero component of the numerator; (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted; (f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient;
and (g) assigning the final remainder as the last subsequent intermediate remainder r(i+1 ) determined at step (c).
In accordance with yet another aspect of the present invention, there is provided a computer-readable medium having computer executable instructions for performing binary integer division to obtain a anal quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the instructions including the steps of: (a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders; whereby the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d:
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
In accordance with still yet another aspect of the present invention, there is provided a computer program product in a computer usable medium, for performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the product having instructions for performing the steps of: (a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d; (b) padding the quotient value q(i) to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); (c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d); (d) padding the subsequent intermediate remainder r(i+1) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator; (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted; (f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient;
and (g) assigning the final remainder as the last subsequent intermediate remainder r(i+1 ) determined at step (c).
Brief Description of Drawings Fig. 1 is an illustrative representation of binary integer division shown in a long division format according to an embodiment of the present invention;
CA9-2003-0101 q.
Fig. 2A to 2C are illustrative examples of binary integer division in long division formats according to an embodiment of the present invention;
Figs. 3 and 4 are flow diagrams of a binary integer division method according to exemplary embodiments of the present invention; and Fig. 5 is a diagram of a computer system useful in implementing the embodiments of the invention.
Detailed Descriution In the field of binary integer division, the computation of a "floor" (2N * n /d) is typically involved in the division process, where N is the number ofbits in a computer system's native integer arithmetic, (e.g. 32-bit, 64-bit, etc.) and numerator (or dividend) n and denominator (or divisor) d are N-bit integers. The floor (2N * n /d) cannot generally be evaluated directly using the computer system's multiply and divide instructions, since the numerator (2~ * n) is generally a 2N-bit integer.
In these situations (termed 2N-bit/ N-bit division) performance can be compromised since a general 2N-bit numerator is assumed and the calculations are implemented in terms of single-bit operations, rather than N-bit operations.
An illustrative representation (in long division format) of binary integer division 100 according to an embodiment of the present invention is shown in Fig. 1. A
numerator (or dividend) 101 is a 2N-bit binary integer comprised of a N-bit binary integer component n 102 and a zero component 104 also of N-bits in length (i.e., an N-bit string of binary 0's).
A denominator (or divisor) d 106 of the binary integer division 100 is also N-bits in length.
The binary integer division 100 generates a final quotient Q I0~ and a final remainder REM 1 I 1. The final quotient Q 108 is obtained from a plurality of quotient values q(i) in a manner described below.
The final remainder REM 111, is determined by the final value of remainder r(i) in a manner described below. The following mathematical notations are used to represent N-bit binary integer arithmetic operations:
addition [+], subtraction [-], multiplication [*], division [/], bitwise or [OR], and bit left shift [«].
As a general overview, the first partial remainder r(0) is initially assigned to be equal to the value of the numerator component n 102 (r(0)=n). The first quotiient value q(0) is initially assigned a value calculated by dividing the first partial remainder r(0) by the denominator d 106 (q(0) _ r(0)[/]d=n[/]d). The first quotient value q(0) becomes the leftmost part of the final quotient Q 108 when q(i) is non-zero (the handling of zero quotient values will be described below). The next partial remainder r(i+1) (shown as expression 112 in Fig. 1) is determined by subtracting from the previous partial remainder r(i) a product of the quotient value q(i) and the denominator d 106 (r(i+1)=r(i)[-] (q(i) [*] d). Therefore, partial remainder r(1) is equal to r(0)[-](q(0)[*]d)= n[-](n[/]d [*] d).
At this stage, the number of bits (rem_bits) in the calculated partial remainder r(i) is considered for subsequent "padding" operations of the partial remainders and quotient values. In particular, the partial remainders r(i) are padded (i.e., binary zeros are appended) such that the number of bits in the partial remainder r(i) is maintained at N-bits in length (i.e., N-rem_bits binary 0's are appended to the r(i)). The padding of binary zeros to the partial remainder is notionally equivalent to "removing" or "bringing down" (shown illustratively with the dotted arrow from component 104 in Fig. 1 ) binary 0°s from the binary zero component 104 of the numerator 101. The binary integer division 100 is considered complete when all of the binary 0's in the zero component 104 are "exhausted" (i.e. all 0's have been removed or brought down) and the final remainder is computed. Further, at each stage {of determining r(i) and q(i)), the quotient values are also "padded" to the left (i.e., N-rem_bits binary 0's are inserted to the front of the calculated q(i)), which is ultimately concatenated with the other quotient values to form the final quotient Q 108. Details of padding partial remainders and quotient values are provided below in conjunction with the description of a specific division example.
Figs. 2A, 2B and 2C represent an illustrative example binary integer division according to an embodiment of the present invention where N=4, the numerator, (2N*n), is n (1101) followed by N zero bits (0000) (24* 13=208 in decimal or 11010000 in binary). The denominator d is 1100 in binary (12 in decimal). The initial partial remainder r(0) is set equal to n, r(0) = n. The first partial quotient q(0) is calculated by dividing r(0) by the denominator (i.e., q{0)=r(0)[/]d=1101 [/] 1100=1).
Binary 1 {i.e., q(0) becomes the first part of the quotient Q. The remainder r(1) is calculated where IS r(I) = r(0) - q(0) [*] d therefore r(1)=I 101-1 [*] I I00=1. The rennainder, r(1), is then padded to the right with the appropriate number of zeros removed from the zero component, so that the remainder r(1) is still of N-bits in length (N-rem_bits = 4 -1 = 3) such that r(1) =
1000. A new partial quotient q(1) is calculated, q(1)=r(1)[/]d=1000[/ ] 1 I00 which is equal to G. Since q(1) is equal to 0, the sub-process is executed using r(1)=1000 and d=1 I00 to calculate a new partial quotient and remainder.
The sub-process continues the division operation by using (N+1)-bit/N-bit division to calculate a new quotient and remainder.
Now referring to Fig. 2C, the computation of (N+1 )-bit/N-bit division is executed by bringing down the remaining 0 of the zero component and padding it to r(1), so that it now equals 10000 (N+1 bits). The remainder r(1) is used as the numerator value in the (N+1)-bit/N-bit division operation.
Since N+1-bit division is not directly possible on N-bit hardware, the sub-process is used. Let n'=1000 be the leftmost N bits of r(1). The sub-process then computes r(1)/d=2*n'/d as follows.
A working value, d2, is calculated by dividing the denominator, d, by two, so that d2=d[/]2=1100[/] 10=110 (binary) =1212=6 (decimal). The least significant bit of d is checked.
Since this bit is 0, d is divisible by two. In this case, the new partial quotient q(1) is recalculated by dividing the numerator n' by the working value d2 (q( 1)= n'/d2=1000[/]
110=1). Since the value of q(1) does not equal zero, a new remainder r(2) is calculated by the numerator n', minus working value d2, multiplied by two(r(2)=2(n[-]d2)=10[*](1000[-] 110)=100). Therefore the new remainder r(2) is equal to 100. The partial quotient q(1) is padded to the left with an equal number of zeros removed from the zero component prior to the (N+1)-bitJN-bit division (3) such that q(1) is equal to 0001. The value of quotient Q is the concatenation of q(0) =l and q (1)=0001 therefore Q=10001(binary) = 17 (decimal). The last remainder value r(2) is the final remainder REM=100 (binary) = 4 (decimal).
Figs. 3 and 4 represent flow diagrams illustrating a method (200 and 300) of performing binary integer division according t~ an exemplary embodiment of the present invention. In the illustrative example shown in Figs. 2A, 2B and 2C, the calculation of the final quotient was performed by concatenation of the values of q(i) and the final rez~ainder as determined by the last value of r(i). In reference to Figs. 3 and 4, the quotient Q is computed by concatenation of the value of the partial quotient b, and remainder REM is calculated directly with binary operations. At step 202 working values of the process 200/300 are initialized, REM = 0, Q = 0, nzeros = Q and z = N, where REM is the remainder, Q is the quotient, nzeros is a counter for the number of zeros brought down during the last operation from the zero component, and ~: is the number of zeros remaining to be brought down from the zero component. At step 204, it is determined whether remainder is greater than or equal to denominator d (REM >_ d) or if there are no more zeros to be brought down from the zero component, (z = 0).
If step 204 is true, (i.e. REM >_ d and z =0) at step 206 the partial quotient, is set equal to the remainder divided by the denominator ( b = REM[/]d). The quotient is then shifted to the left by the number of zeros brought down from the zero component during the last operation (Q shifted left by nzeros). The partial quotient is inserted into the quotient Q by a bitwise OR
of quotient and partial quotient (Q = Q[~R]b). At step 210 the zero status is once again determined.
If there are no more remaining zeros to be brought down (z = 0), then the final result of quotient (Q) and remainder (REM) are displayed at step 212 terminating the process. If there are zeros remaining to be brought down (z ~0), at step 211 the remainder is set equal to the remainder minus the partial quotient multiplied by denominator (REM=REM[-](b[*]d). The process then continues at step 204.
If step 204 is false, (i.e. REM < d or z r0), there are still zeros to be brought down however (N+1)-bitlN-bit division is required so that progress may continue: at step 208. Now referring to Fig.
4, a sub-process 300 calculates the (N+1)-bit/N-bit division 2*n'/d, where the current value of REM
is used as the numerator n' in the sub-process 300 (n'=REM). At step 302, a working value, d2, is CA9-2003-0101 g assigned equal to the denominator divided by two (d2= d [/] 2) which can be performed by a right shift operation of the denominator in binary arithmetic.
At step 304 it is then determined if the denominator is divisible by two by determining if the least significant bit of the denominator is zero. If the denominator is divisible by two, then the working value is exact. At step 306, since there is no remainder the partial quotient is set to equal to the numerator divided by the working value (b= n' [/] d2).
At step 308, if the partial quotient is equal to 0 (b = 0), then the remainder is set equal the numerator multiplied by two (IZEM = 2 [*] n') at step 312. If the; partial quotient is not equal to 0 b # 0) at step 308, then the remainder is set to the numerator minus the working value and then multiplied by two ( REM = 2 [*] ( n' [-] d2 ) at step 314.
If the denominator is not divisible by two at step 304, therefore d/2 is not an integer, it is then determined whether the numerator n' is less than or equal to the working value d2 (n' <= d2) at step 310.
If the numerator n' is less than or equal to the working value (n' < =d2) at step 310, then the numerator is less than the denominator divided by two ( n'< d/2) therefore the partial quotient is set equal to zero (b = 0) and the remainder is set equal to the numerator multiplied by two (REM = 2 (*) n') at step 316. However, if n' is greater than the working value ( n'>
d2) at step 310, the numerator is greater than or equal to the working value plu s one, which is greater than the denominator divided by two ( n' >_ d2 +1 > d/2 ), so that the numerator multiplied by two, divided by the denominator, is greater than one ( 2*n'/ d > 1) and partial quotient is equal to one (b = 1).
Therefore at step 318, the remainder is then equal to the product of the numerator multiplied by two, minus the denominator which is equal to the numerator minus working value multiplied by two REM = 2*n'-d = 2* (n - d/2)). However, since the denominator divided by two ( d/2 ) is not an integer, the denominator divided by two is equal to the working value plus one-half ( d/2 = d2 + 1/2), so that the remainder is equal to the product of the numerator minus the working value minus one-half multiplied by two, which is also equal to the numerator minus the working value multiplied by two minus one (REM = 2 * (n - d2 -1/2) = 2 * (n - d2) -1=2 [*] (n' [-] d2) [-]
1). The new values of the partial quotient and the remainder are returned to the main process to continue the division.
Referring again to Fig. 3, at step 214, the number of zeros last brought down is incremented by one (nzeros = nzeros[+] 1) and the number of zeros remaining is decreased by one ( z = z [-] 1) as the sub-process implicitly brings down one zero. The quotient is shifted left by the number of zeros last brought down to make room for the partial quotient (q = q + nzeros). The partial quotient is then inserted into the quotient by a bitwise OR operation (Q = Q [OR] b). The number of leading zeros in the remainder indicates the number of bits by which the process has progressed. The number of zeros last brought down is then set equal to the minimum of the number of leading zeros in the remainder, rem_bits, and the zeros remaining to be brought down (nzeros = MIN
(rem_bits, z)). The remainder is shifted left by number of remaining zeros last brought down (REM
=REM [«] nzeros) and the number of zeros remaining to be brought down is decrernented by the number of zeros last brought down ( z = z[-) nzeros ). The process then restarts at step 204 until the final value for Q and REM are displayed at step 212.
An example of the exemplary embodiment of binary integer division described in Fig. 3 and Fig.4 implemented in C programming language is provided below. The TNDIV
function implements the main process 200 while N2DIV implements the sub-process 300.
TNDIV (n,d) function #define MIN(x,y) (x<y?x:y}
extern unsigned int CNTLZ (UNSIGNED);
/* Returns number of leading zeros */
extern UNSIGNED N2DIV (UNSIGNED, UNSIGNED, UNSIG1VED *);
/* See next process */
UNSIGNED TNDIV (UNSIGNED n, UNSIGNED d) /* Returns floor (2**N * n / d), where n < d and N is the word length. */
UNSIGNED rem, b, q, z, rem_bits, b_bits, nzeros, rem2;
rem = n; /* Current remainder */
q = 0; /* Quotient so far */
z = N; /* Number of zeros remaining to be brought down. */
nzeros = 0; /* Number of zeros last brought down. */
while ( 1 ) {
if ((rem >= d) I, (z = = 0)) /*Either there are no more zeros to be brought down, in which case this is the last iteration, or we can make progress using N-bit/N-bit division. */
/* Do N-bit divide. */
b = rem/d;
/* Make room for partial quotient and insert it, */
q «= nzeros;
q I= b;
/* Done if no more zeros to bring down. */
if (z == 0) break;
/* Compute new remainder */
rem -= d*b;
else {
I* There are still zeros to be brought down, and we need to use (N+1)-bit/N-bit division to make progress. */
/* Do (N+1)-bitlN-bit divide of 2*rem/d. This implicitly brings down one more zero. */
b = N2DIV (rem, d, &rem);
nzeros += l;
z -= l;
/* Make room for partial quotient and insert it. */
q «= nzeros;
q ~=b~
/* Determine number of leading zeros in remainder, which is the number of bits we have progressed by. */
rem bits = CNTLZ (rem) /* Determine number of zeros to bring down and bring there down. */
nzeros = MIN (rem bits, z);
rem «= nzeros;
z -= nzeros;
return q;
N2DIV functi~n UNSIGNED N2DIV (UNSIGNED n, UNSIGNED d, UNSIGNED *r) /* Returns floor (2 * n / d) . Sets r to remainder. Must have n < d. */
{
UNSIGNED d2, q;
d2 = d » l; /* d2 = dl2 */
if (!(d & 1)) {
/* d is divisible by 2, so d2 is exact */
q=n/d2;
if (q = = 0) {
*r=2*n;
else {
*r =2 ~ (n - d2);
else {
/* d is not divisible by 2 */
if (n <= d2) {
q=0;
*r = 2 * n;
}
else {
q = 1;
~r = 2 ~' (n - d2) -1;
}
return q;
{
The hardware elements of a computer system used to implement the present invention are shown in Fig. 5. A central processing unit (CPU) 400 provides main processing functionality. A memory 410 is coupled to CPU 400 for providing operational storage of programs and data. Memory 410 may comprise, for example, random. access memory (RAM) or read only memory (R~M). l~Ton-volatile storage of, for example, data files and programs is provided by a storage 420 that may comprise, for example, disk storage. Both memory 410 and storage 420 comprise a computer useable medium that may store computer program products in the form of computer readable program code.
User input and output is provided by an inputJoutput (I/O) facility 430 The I/O facility 430 may include, for example, a graphical display, a mouse and/or a keyboard.
Field of Invention The present invention relates to the field of mathematical processing in computer systems and more particularly to methods of performing binary integer division.
Back round The execution of binary integer division in many computer systems requires considerably more processing cycles than other operations such as binary integer multiplication. As a result, methods have been developed for computing binary integer division by using low-cost operations, in terms of processor cycles and execution time required, such as shift, multiply and add rather than hardware based integer division. Binary integer division methods are used in compilers for generating program code used to evaluate integer divisions, for example, in programming loops with compile-time known loop-invariant divisors (i.e., the divisor is the same for each division).
Programs resulting from implementation of these methods typically run faster than programs using the native hardware binary integer division, thus improving execution time and overall performance.
However, when the divisor of a binary integer division operation used in a program loop is Ioop-invariant, but its value is not known at compile time, changes to traditional processing methods are required.
Surnmary of Invention In accordance with one aspect of the present invention, there is provided a method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method including: (a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders; wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d; q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
In accordance with another aspect of the present invention, there is provided a method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method including: (a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d; (6) padding the quotient value q(i) to the left with binary 0's equal to N less the number of bits in the intermediate Ct~9-2003-0101 2 remainder r(i); (c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d); (d) padding the subsequent intermediate remainder r(i+1 ) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from tale zero component of the numerator; (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted; (f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient;
and (g) assigning the final remainder as the last subsequent intermediate remainder r(i+1 ) determined at step (c).
In accordance with yet another aspect of the present invention, there is provided a computer-readable medium having computer executable instructions for performing binary integer division to obtain a anal quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the instructions including the steps of: (a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders; whereby the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d:
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
In accordance with still yet another aspect of the present invention, there is provided a computer program product in a computer usable medium, for performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the product having instructions for performing the steps of: (a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d; (b) padding the quotient value q(i) to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); (c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d); (d) padding the subsequent intermediate remainder r(i+1) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator; (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted; (f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient;
and (g) assigning the final remainder as the last subsequent intermediate remainder r(i+1 ) determined at step (c).
Brief Description of Drawings Fig. 1 is an illustrative representation of binary integer division shown in a long division format according to an embodiment of the present invention;
CA9-2003-0101 q.
Fig. 2A to 2C are illustrative examples of binary integer division in long division formats according to an embodiment of the present invention;
Figs. 3 and 4 are flow diagrams of a binary integer division method according to exemplary embodiments of the present invention; and Fig. 5 is a diagram of a computer system useful in implementing the embodiments of the invention.
Detailed Descriution In the field of binary integer division, the computation of a "floor" (2N * n /d) is typically involved in the division process, where N is the number ofbits in a computer system's native integer arithmetic, (e.g. 32-bit, 64-bit, etc.) and numerator (or dividend) n and denominator (or divisor) d are N-bit integers. The floor (2N * n /d) cannot generally be evaluated directly using the computer system's multiply and divide instructions, since the numerator (2~ * n) is generally a 2N-bit integer.
In these situations (termed 2N-bit/ N-bit division) performance can be compromised since a general 2N-bit numerator is assumed and the calculations are implemented in terms of single-bit operations, rather than N-bit operations.
An illustrative representation (in long division format) of binary integer division 100 according to an embodiment of the present invention is shown in Fig. 1. A
numerator (or dividend) 101 is a 2N-bit binary integer comprised of a N-bit binary integer component n 102 and a zero component 104 also of N-bits in length (i.e., an N-bit string of binary 0's).
A denominator (or divisor) d 106 of the binary integer division 100 is also N-bits in length.
The binary integer division 100 generates a final quotient Q I0~ and a final remainder REM 1 I 1. The final quotient Q 108 is obtained from a plurality of quotient values q(i) in a manner described below.
The final remainder REM 111, is determined by the final value of remainder r(i) in a manner described below. The following mathematical notations are used to represent N-bit binary integer arithmetic operations:
addition [+], subtraction [-], multiplication [*], division [/], bitwise or [OR], and bit left shift [«].
As a general overview, the first partial remainder r(0) is initially assigned to be equal to the value of the numerator component n 102 (r(0)=n). The first quotiient value q(0) is initially assigned a value calculated by dividing the first partial remainder r(0) by the denominator d 106 (q(0) _ r(0)[/]d=n[/]d). The first quotient value q(0) becomes the leftmost part of the final quotient Q 108 when q(i) is non-zero (the handling of zero quotient values will be described below). The next partial remainder r(i+1) (shown as expression 112 in Fig. 1) is determined by subtracting from the previous partial remainder r(i) a product of the quotient value q(i) and the denominator d 106 (r(i+1)=r(i)[-] (q(i) [*] d). Therefore, partial remainder r(1) is equal to r(0)[-](q(0)[*]d)= n[-](n[/]d [*] d).
At this stage, the number of bits (rem_bits) in the calculated partial remainder r(i) is considered for subsequent "padding" operations of the partial remainders and quotient values. In particular, the partial remainders r(i) are padded (i.e., binary zeros are appended) such that the number of bits in the partial remainder r(i) is maintained at N-bits in length (i.e., N-rem_bits binary 0's are appended to the r(i)). The padding of binary zeros to the partial remainder is notionally equivalent to "removing" or "bringing down" (shown illustratively with the dotted arrow from component 104 in Fig. 1 ) binary 0°s from the binary zero component 104 of the numerator 101. The binary integer division 100 is considered complete when all of the binary 0's in the zero component 104 are "exhausted" (i.e. all 0's have been removed or brought down) and the final remainder is computed. Further, at each stage {of determining r(i) and q(i)), the quotient values are also "padded" to the left (i.e., N-rem_bits binary 0's are inserted to the front of the calculated q(i)), which is ultimately concatenated with the other quotient values to form the final quotient Q 108. Details of padding partial remainders and quotient values are provided below in conjunction with the description of a specific division example.
Figs. 2A, 2B and 2C represent an illustrative example binary integer division according to an embodiment of the present invention where N=4, the numerator, (2N*n), is n (1101) followed by N zero bits (0000) (24* 13=208 in decimal or 11010000 in binary). The denominator d is 1100 in binary (12 in decimal). The initial partial remainder r(0) is set equal to n, r(0) = n. The first partial quotient q(0) is calculated by dividing r(0) by the denominator (i.e., q{0)=r(0)[/]d=1101 [/] 1100=1).
Binary 1 {i.e., q(0) becomes the first part of the quotient Q. The remainder r(1) is calculated where IS r(I) = r(0) - q(0) [*] d therefore r(1)=I 101-1 [*] I I00=1. The rennainder, r(1), is then padded to the right with the appropriate number of zeros removed from the zero component, so that the remainder r(1) is still of N-bits in length (N-rem_bits = 4 -1 = 3) such that r(1) =
1000. A new partial quotient q(1) is calculated, q(1)=r(1)[/]d=1000[/ ] 1 I00 which is equal to G. Since q(1) is equal to 0, the sub-process is executed using r(1)=1000 and d=1 I00 to calculate a new partial quotient and remainder.
The sub-process continues the division operation by using (N+1)-bit/N-bit division to calculate a new quotient and remainder.
Now referring to Fig. 2C, the computation of (N+1 )-bit/N-bit division is executed by bringing down the remaining 0 of the zero component and padding it to r(1), so that it now equals 10000 (N+1 bits). The remainder r(1) is used as the numerator value in the (N+1)-bit/N-bit division operation.
Since N+1-bit division is not directly possible on N-bit hardware, the sub-process is used. Let n'=1000 be the leftmost N bits of r(1). The sub-process then computes r(1)/d=2*n'/d as follows.
A working value, d2, is calculated by dividing the denominator, d, by two, so that d2=d[/]2=1100[/] 10=110 (binary) =1212=6 (decimal). The least significant bit of d is checked.
Since this bit is 0, d is divisible by two. In this case, the new partial quotient q(1) is recalculated by dividing the numerator n' by the working value d2 (q( 1)= n'/d2=1000[/]
110=1). Since the value of q(1) does not equal zero, a new remainder r(2) is calculated by the numerator n', minus working value d2, multiplied by two(r(2)=2(n[-]d2)=10[*](1000[-] 110)=100). Therefore the new remainder r(2) is equal to 100. The partial quotient q(1) is padded to the left with an equal number of zeros removed from the zero component prior to the (N+1)-bitJN-bit division (3) such that q(1) is equal to 0001. The value of quotient Q is the concatenation of q(0) =l and q (1)=0001 therefore Q=10001(binary) = 17 (decimal). The last remainder value r(2) is the final remainder REM=100 (binary) = 4 (decimal).
Figs. 3 and 4 represent flow diagrams illustrating a method (200 and 300) of performing binary integer division according t~ an exemplary embodiment of the present invention. In the illustrative example shown in Figs. 2A, 2B and 2C, the calculation of the final quotient was performed by concatenation of the values of q(i) and the final rez~ainder as determined by the last value of r(i). In reference to Figs. 3 and 4, the quotient Q is computed by concatenation of the value of the partial quotient b, and remainder REM is calculated directly with binary operations. At step 202 working values of the process 200/300 are initialized, REM = 0, Q = 0, nzeros = Q and z = N, where REM is the remainder, Q is the quotient, nzeros is a counter for the number of zeros brought down during the last operation from the zero component, and ~: is the number of zeros remaining to be brought down from the zero component. At step 204, it is determined whether remainder is greater than or equal to denominator d (REM >_ d) or if there are no more zeros to be brought down from the zero component, (z = 0).
If step 204 is true, (i.e. REM >_ d and z =0) at step 206 the partial quotient, is set equal to the remainder divided by the denominator ( b = REM[/]d). The quotient is then shifted to the left by the number of zeros brought down from the zero component during the last operation (Q shifted left by nzeros). The partial quotient is inserted into the quotient Q by a bitwise OR
of quotient and partial quotient (Q = Q[~R]b). At step 210 the zero status is once again determined.
If there are no more remaining zeros to be brought down (z = 0), then the final result of quotient (Q) and remainder (REM) are displayed at step 212 terminating the process. If there are zeros remaining to be brought down (z ~0), at step 211 the remainder is set equal to the remainder minus the partial quotient multiplied by denominator (REM=REM[-](b[*]d). The process then continues at step 204.
If step 204 is false, (i.e. REM < d or z r0), there are still zeros to be brought down however (N+1)-bitlN-bit division is required so that progress may continue: at step 208. Now referring to Fig.
4, a sub-process 300 calculates the (N+1)-bit/N-bit division 2*n'/d, where the current value of REM
is used as the numerator n' in the sub-process 300 (n'=REM). At step 302, a working value, d2, is CA9-2003-0101 g assigned equal to the denominator divided by two (d2= d [/] 2) which can be performed by a right shift operation of the denominator in binary arithmetic.
At step 304 it is then determined if the denominator is divisible by two by determining if the least significant bit of the denominator is zero. If the denominator is divisible by two, then the working value is exact. At step 306, since there is no remainder the partial quotient is set to equal to the numerator divided by the working value (b= n' [/] d2).
At step 308, if the partial quotient is equal to 0 (b = 0), then the remainder is set equal the numerator multiplied by two (IZEM = 2 [*] n') at step 312. If the; partial quotient is not equal to 0 b # 0) at step 308, then the remainder is set to the numerator minus the working value and then multiplied by two ( REM = 2 [*] ( n' [-] d2 ) at step 314.
If the denominator is not divisible by two at step 304, therefore d/2 is not an integer, it is then determined whether the numerator n' is less than or equal to the working value d2 (n' <= d2) at step 310.
If the numerator n' is less than or equal to the working value (n' < =d2) at step 310, then the numerator is less than the denominator divided by two ( n'< d/2) therefore the partial quotient is set equal to zero (b = 0) and the remainder is set equal to the numerator multiplied by two (REM = 2 (*) n') at step 316. However, if n' is greater than the working value ( n'>
d2) at step 310, the numerator is greater than or equal to the working value plu s one, which is greater than the denominator divided by two ( n' >_ d2 +1 > d/2 ), so that the numerator multiplied by two, divided by the denominator, is greater than one ( 2*n'/ d > 1) and partial quotient is equal to one (b = 1).
Therefore at step 318, the remainder is then equal to the product of the numerator multiplied by two, minus the denominator which is equal to the numerator minus working value multiplied by two REM = 2*n'-d = 2* (n - d/2)). However, since the denominator divided by two ( d/2 ) is not an integer, the denominator divided by two is equal to the working value plus one-half ( d/2 = d2 + 1/2), so that the remainder is equal to the product of the numerator minus the working value minus one-half multiplied by two, which is also equal to the numerator minus the working value multiplied by two minus one (REM = 2 * (n - d2 -1/2) = 2 * (n - d2) -1=2 [*] (n' [-] d2) [-]
1). The new values of the partial quotient and the remainder are returned to the main process to continue the division.
Referring again to Fig. 3, at step 214, the number of zeros last brought down is incremented by one (nzeros = nzeros[+] 1) and the number of zeros remaining is decreased by one ( z = z [-] 1) as the sub-process implicitly brings down one zero. The quotient is shifted left by the number of zeros last brought down to make room for the partial quotient (q = q + nzeros). The partial quotient is then inserted into the quotient by a bitwise OR operation (Q = Q [OR] b). The number of leading zeros in the remainder indicates the number of bits by which the process has progressed. The number of zeros last brought down is then set equal to the minimum of the number of leading zeros in the remainder, rem_bits, and the zeros remaining to be brought down (nzeros = MIN
(rem_bits, z)). The remainder is shifted left by number of remaining zeros last brought down (REM
=REM [«] nzeros) and the number of zeros remaining to be brought down is decrernented by the number of zeros last brought down ( z = z[-) nzeros ). The process then restarts at step 204 until the final value for Q and REM are displayed at step 212.
An example of the exemplary embodiment of binary integer division described in Fig. 3 and Fig.4 implemented in C programming language is provided below. The TNDIV
function implements the main process 200 while N2DIV implements the sub-process 300.
TNDIV (n,d) function #define MIN(x,y) (x<y?x:y}
extern unsigned int CNTLZ (UNSIGNED);
/* Returns number of leading zeros */
extern UNSIGNED N2DIV (UNSIGNED, UNSIGNED, UNSIG1VED *);
/* See next process */
UNSIGNED TNDIV (UNSIGNED n, UNSIGNED d) /* Returns floor (2**N * n / d), where n < d and N is the word length. */
UNSIGNED rem, b, q, z, rem_bits, b_bits, nzeros, rem2;
rem = n; /* Current remainder */
q = 0; /* Quotient so far */
z = N; /* Number of zeros remaining to be brought down. */
nzeros = 0; /* Number of zeros last brought down. */
while ( 1 ) {
if ((rem >= d) I, (z = = 0)) /*Either there are no more zeros to be brought down, in which case this is the last iteration, or we can make progress using N-bit/N-bit division. */
/* Do N-bit divide. */
b = rem/d;
/* Make room for partial quotient and insert it, */
q «= nzeros;
q I= b;
/* Done if no more zeros to bring down. */
if (z == 0) break;
/* Compute new remainder */
rem -= d*b;
else {
I* There are still zeros to be brought down, and we need to use (N+1)-bit/N-bit division to make progress. */
/* Do (N+1)-bitlN-bit divide of 2*rem/d. This implicitly brings down one more zero. */
b = N2DIV (rem, d, &rem);
nzeros += l;
z -= l;
/* Make room for partial quotient and insert it. */
q «= nzeros;
q ~=b~
/* Determine number of leading zeros in remainder, which is the number of bits we have progressed by. */
rem bits = CNTLZ (rem) /* Determine number of zeros to bring down and bring there down. */
nzeros = MIN (rem bits, z);
rem «= nzeros;
z -= nzeros;
return q;
N2DIV functi~n UNSIGNED N2DIV (UNSIGNED n, UNSIGNED d, UNSIGNED *r) /* Returns floor (2 * n / d) . Sets r to remainder. Must have n < d. */
{
UNSIGNED d2, q;
d2 = d » l; /* d2 = dl2 */
if (!(d & 1)) {
/* d is divisible by 2, so d2 is exact */
q=n/d2;
if (q = = 0) {
*r=2*n;
else {
*r =2 ~ (n - d2);
else {
/* d is not divisible by 2 */
if (n <= d2) {
q=0;
*r = 2 * n;
}
else {
q = 1;
~r = 2 ~' (n - d2) -1;
}
return q;
{
The hardware elements of a computer system used to implement the present invention are shown in Fig. 5. A central processing unit (CPU) 400 provides main processing functionality. A memory 410 is coupled to CPU 400 for providing operational storage of programs and data. Memory 410 may comprise, for example, random. access memory (RAM) or read only memory (R~M). l~Ton-volatile storage of, for example, data files and programs is provided by a storage 420 that may comprise, for example, disk storage. Both memory 410 and storage 420 comprise a computer useable medium that may store computer program products in the form of computer readable program code.
User input and output is provided by an inputJoutput (I/O) facility 430 The I/O facility 430 may include, for example, a graphical display, a mouse and/or a keyboard.
Claims (12)
1. A method of performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method comprising:
(a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders;
wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d;
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r{i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
(a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders;
wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d;
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r{i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
2. The method of claim 1, wherein when one of the quotient values q(i) is equal to zero and the denominator is divisible by two, the method further comprising:
defining a recalculated quotient value q(i) as n'[/]d2;
determining the intermediate remainder as r(i~2[*]n' when the recalculated quotient value q(i) is equal to zero; and determining the intermediate remainder as r(i~2[*](n' [-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining a recalculated quotient value q(i) as n'[/]d2;
determining the intermediate remainder as r(i~2[*]n' when the recalculated quotient value q(i) is equal to zero; and determining the intermediate remainder as r(i~2[*](n' [-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
3. The method of claim 1, wherein when one of the quotient values q(i) is equal to zero and the denominator is not divisible by two, the method further comprising:
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
4. A method of performing binary integer division to obtain a final quotient and a anal remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the method comprising:
(a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=-n and q(0)=r(0)[/]d;
(b) padding the quotient value q(i) to the left with binary 0's equal to N
less the number of bits in the intermediate remainder r(i);
(c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d);
(d) padding the subsequent intermediate remainder r(i+1) to the right with binary 4's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator;
(e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted;
(f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient; and (g) assigning the final remainder as the subsequent intermediate remainder r(i+1) determined at step (c).
(a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=-n and q(0)=r(0)[/]d;
(b) padding the quotient value q(i) to the left with binary 0's equal to N
less the number of bits in the intermediate remainder r(i);
(c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d);
(d) padding the subsequent intermediate remainder r(i+1) to the right with binary 4's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator;
(e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted;
(f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient; and (g) assigning the final remainder as the subsequent intermediate remainder r(i+1) determined at step (c).
5. The method of claim 4, wherein when one of the quotient values q(i) determined at step (a) is equal to zero and the denominator is divisible by two, the method further comprising:
recalculating the quotient value q(i) as n' [/]d2;
calculating the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and calculating the intermediate remainder as r(i)=,2[*](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
recalculating the quotient value q(i) as n' [/]d2;
calculating the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and calculating the intermediate remainder as r(i)=,2[*](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
6. The method of claim 4, wherein when one of the quotient values q(i) determined at step (a) is equal to zero and the denominator is not divisible by two, the method further comprising:
defining the quotient value q(i) as binary 0 and. the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining the quotient value q(i) as binary 0 and. the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
7. A computer-readable medium having computer executable instructions for performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the instructions comprising the steps of:
(a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders;
wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d:
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
(a) delivering the denominator and intermediate remainders to a computer system operable to determine quotient values; and (b) delivering the denominator and quotient values to a computer system operable to determine the intermediate remainders;
wherein the quotient values q(i) and the intermediate remainders r(i) are generated by performing the following assignments iteratively (i=i+1) until the binary 0's of the zero component of the numerator are exhausted, in which r(0) is initially set equal to n and q(0) is initially set equal to r(0)[/]d:
q(i)=r(i)[/]d padded to the left with binary 0's equal to N less the number of bits in the intermediate remainder r(i); and r(i+1)=r(i)[-](q(i)[*]d) padded to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, wherein the final quotient is represented by a concatenation of the quotient values q(i) and the final remainder is represented by the value of r(i).
8. The computer-readable medium of claim 7, wherein when one of the quotient values q(i) is equal to zero and the denominator is divisible by two, the instructions further comprise the steps of:
defining a recalculated quotient value q(i) as n'[/]d2;
determining the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and determining the intermediate remainder as r(i)=2[n'](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining a recalculated quotient value q(i) as n'[/]d2;
determining the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and determining the intermediate remainder as r(i)=2[n'](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
9. The computer-readable medium of claim 7, wherein when one of the quotient values q(i) is equal to zero and the denominator is not divisible by two, the instructions further comprise the steps of:
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
10. A computer program product in a computer usable medium, for performing binary integer division to obtain a final quotient and a final remainder from an N-bit denominator d and a 2N-bit numerator comprising a binary integer component n of N-bits and a zero component of N-bits of binary 0's, the computer program product comprising instructions for performing the steps of:
(a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d;
(b) padding the quotient value q(i) to the left with binary 0's equal to N
less the number of bits in the intermediate remainder r(i);
(c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d);
(d) padding the subsequent intermediate remainder r(i+1) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted;
(f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient; and (g) assigning the final remainder as the subsequent intermediate remainder r(i+1) determined at step (c).
(a) calculating a quotient value q(i) as r(i)[/]d, wherein r(i) is an intermediate remainder and r(0)=n and q(0)=r(0)[/]d;
(b) padding the quotient value q(i) to the left with binary 0's equal to N
less the number of bits in the intermediate remainder r(i);
(c) calculating a subsequent intermediate remainder r(i+1) as r(i)[-](q(i)[*]d);
(d) padding the subsequent intermediate remainder r(i+1) to the right with binary 0's equal to N less the number of bits in the intermediate remainder r(i) removed from the zero component of the numerator, (e) repeating steps (a)-(d) until the binary 0's of the zero component of the numerator are exhausted;
(f) concatenating the quotient values q(i) obtained from step (a) to form the final quotient; and (g) assigning the final remainder as the subsequent intermediate remainder r(i+1) determined at step (c).
11. The computer program product of claim 10, wherein when one of the quotient values q(i) determined at step (a) is equal to zero and the denominator is divisible by two, the product further comprising instructions for performing the steps of:
recalculating the quotient value q(i) as n' [/]d2;
calculating the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and calculating the intermediate remainder as r(i)=2[*](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
recalculating the quotient value q(i) as n' [/]d2;
calculating the intermediate remainder as r(i)=2[*]n' when the recalculated quotient value q(i) is equal to zero; and calculating the intermediate remainder as r(i)=2[*](n'[-]d2) when the recalculated quotient value q(i) is not equal to zero;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
12. The computer program product of claim 10, wherein when one of the quotient values q(i) determined at step (a) is equal to zero and the denominator is not divisible by two, the product further comprising instructions for performing the steps of:
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
defining the quotient value q(i) as binary 0 and the intermediate remainder r(i) as 2[*]n' when n' is less than or equal to d2; and defining the quotient value q(i) as binary 1 and the intermediate remainder r(i) as 2[*](n'[-]d2)[-]1 when n' is greater than d2;
wherein d2 is equal to d[/]2 and n' is the left most N bits of r(i) padded with a binary 0 removed from the zero component of the numerator.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA002453712A CA2453712A1 (en) | 2003-12-17 | 2003-12-17 | Method of performing binary integer division |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA002453712A CA2453712A1 (en) | 2003-12-17 | 2003-12-17 | Method of performing binary integer division |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2453712A1 true CA2453712A1 (en) | 2005-06-17 |
Family
ID=34658592
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA002453712A Abandoned CA2453712A1 (en) | 2003-12-17 | 2003-12-17 | Method of performing binary integer division |
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| Country | Link |
|---|---|
| CA (1) | CA2453712A1 (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2011051769A2 (en) * | 2009-10-26 | 2011-05-05 | Jadavpur University | Method and system for determining a quotient value |
-
2003
- 2003-12-17 CA CA002453712A patent/CA2453712A1/en not_active Abandoned
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2011051769A2 (en) * | 2009-10-26 | 2011-05-05 | Jadavpur University | Method and system for determining a quotient value |
| WO2011051769A3 (en) * | 2009-10-26 | 2011-06-23 | Jadavpur University | Method and system for determining a quotient value |
| US8694573B2 (en) | 2009-10-26 | 2014-04-08 | Jadavpur University | Method and system for determining a quotient value |
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