JP2708526B2 - Binary integer division processing method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Industrial Application Field] The present invention relates to a binary for dividing a dividend and a divisor having a length twice as long as a binary integer handled by a division instruction execution mechanism of a computer. It relates to an integer division processing method.

Division is frequently used in scientific and technological calculations and paperwork calculations by computers. The simplest form of numerical values handled by the computer is a binary integer, but it is desired to increase the number of digits in the dividend and divisor represented by the binary integer to be divided.

[Conventional technology]

Conventionally, a FO executed by a computer with 32 bits per word
In RTRAN programs, division between 31-bit integer types is possible, but division between 63-bit integer types cannot be performed.

[Problems to be solved by the invention]

Conventionally, in a FORTRAN program, etc., it was not possible to divide a 63-bit integer type. To perform such a division, for example, convert an integer type number to a floating point and perform the division using a floating point instruction. After that, there is a problem that it is necessary to perform a process of performing type conversion on the result to an integer type. Therefore, there is a problem that the processing procedure is complicated and the processing is slow.

An object of the present invention is to solve the above problems and to provide means for realizing 63-bit integer type division in a FORTRAN program or the like without using a floating-point instruction, and for accurately obtaining the quotient and remainder of the division. And

[Means for solving the problem]

 FIG. 1 shows a configuration example of the present invention.

In FIG. 1, 10 is a computer including a CPU and a memory, etc., P1 to P3 are processing steps for division by the computer 10,
X is dividend, Y is divisor, d (X) is the number of significant digits of dividend X, d
(Y) represents the number of significant digits of the divisor Y.

The computer 10 has a plurality of registers for holding n-bit data, and has a binary integer division instruction execution mechanism set in these registers.

When dividing the dividend X and the divisor Y, each of which is a binary integer represented in the range of 2n bits, the division is performed by the following processing steps P1 to P3.

In the process P1, the number of significant digits d of the dividend X and the divisor Y is
(X) and d (Y) are obtained. The number of significant digits is the number of bits from the most significant bit taking the value of "1" to the least significant bit. Also, the difference d between the number of significant digits d (X) and the number of significant digits d (Y)
Ask for.

In the process P2, the number of significant digits d (X), the number of significant digits d
From (Y) and the difference d, the following obvious quotient Z is obtained.

If d (Y) = 0, it is a division error.

If d (X) = 0, then Z = 0.

If d <0, Z = 0.

If d = 0, then Z = 0 or 1.

If d (Y) = 1, then Z = X.

In the next processing step P2, n
In order to calculate the quotient Z using the division of bits, the cases are classified according to whether the number of significant digits d (Y) is less than or equal to n-1 or greater than or equal to n. , The dividend X and the divisor Y are shifted to obtain values U and V whose effective digits are n bits or less, and a quotient Z is obtained by the division. And
The obtained quotient Z is corrected by the back calculation.

[Action]

In the present invention, when performing a division on a dividend X and a divisor Y represented within a range of 2n bits by a computer 10 in which one word is n bits, first, the number of significant digits thereof is obtained. With this number of significant digits, a division error or a trivial quotient can be obtained.

Further, by paying attention to the number of significant digits of the divisor Y, the division is performed, and the division is executed in such a manner that a division instruction by an n-bit divisor can be used.

〔Example〕

FIG. 2 shows an example of an application system according to one embodiment of the present invention, FIG. 3 shows an example of processing according to one embodiment of the present invention, and FIG. 4 shows a processing flow of one embodiment of the present invention.

The present invention is implemented, for example, in a system as shown in FIG. In FIG. 2, reference numeral 20 denotes a FORTRAN object program obtained by compiling a program described in the FORTRAN language, reference numeral 21 denotes a parameter list, and reference numeral 22 denotes a 63-bit division function routine provided as a library.

Conventionally, in a FORTRAN language program, division between 63-bit integer types has not been possible, which has been a limitation of language specifications. In this embodiment, division of a 63-bit integer type is possible, and when performing this, the FORTRAN object program 20 creates a parameter list 21 including a dividend X, a divisor Y, and a result area of a 63-bit integer type, respectively. Then, the previously prepared 63-bit division function routine 22 is called.

The 63-bit division function routine 22 executes a 63-bit integer type division through the processing steps shown in FIG. Then, the quotient Z of the division result is set in the result area of the parameter list 21, and the process returns to the calling FORTRAN object program 20.

For example, as shown in FIG. 3A, it is assumed that the division of "I = 2147483648/2" is performed in the FORTRAN object program 20.

The FORTRAN object program 20 is shown in FIG.
A parameter list 21 as shown in (1) is created, the start address is set in a predetermined register Reg, and a 63-bit division function routine 22 is called.

The principle of the operation performed by the 63-bit division function routine 22 is as follows.

 Consider Z = X / Y. Here, it is assumed that X ≧ 0 and Y ≧ 0.

The significant digits of binary integers of X and Y are d (X), d (Y)
And the difference is d. The number of significant digits of Z is d (Z)
And

d = d (X) −d (Y) d ≦ d (Z) ≦ d + 1.

 From this, we can immediately see the following.

If d (Y) = 0, division error d (X) = 0, Z = 0 d <0, Z = 0 d = 0, Z = 0 or 1 d (Y) = 1 in the case of Z = X X = -2 ^63, the normal processing, the division instruction (D instruction, DR instructions, etc.) of the computer 10 can not be used.
Therefore, in the case of X = -2 ⁶³ divides the X / 2 in Y, and the quotient (P), according to the relationship remainder (R), be as follows Z.

If R · 2 ≧ Y, Z = 2P + 1 If R · 2 <Y, Z = 2P The operation is performed using a well-known division instruction (D instruction or DR instruction). is there.

i) Results exception of overflow if (digits is 2 ³¹ or more is produced.) ii) In the case of Y ≧ 2 ³¹ (divisor can not be represented by 4 bytes.) Based on these things, the following (a) , (B).

In this case the case of (a) d (Y) ≦ 31, the divisor Y are, because it is Y <2 ^31, the (i)
You only have to worry about the problem.

1) When d ≦ 30 In this case, since d (Z) ≦ d + 1 ≦ 31, the quotient Z is 4
Value within bytes.

Therefore, Z = X / Y2 may be calculated.

(However, Y2 is the lower 4 bytes of Y) when the case of 2) d ≧ 31 X = X1 · 2 31 + X2 (0 ≦ X2 <2 31), Z = X / Y = X1 · 2 31 / Y + X2 / Y X1 = the P · Y + and R (0 ≦ R <Y) , Z = (P · Y + R) · 2 31 / Y + X2 / Y = P · 2 31 + (R · 2 31 + X) / Y , where, 0 ≦ R <Y, from ^{0 ≦ X2 <2 31, R} · 2 31 + X2 <(R
^{+1) 2 31 ≦ Y · 2} 31 words, with ^{(R · 2 31 + X2)} / Y,
No overflow occurs.

In addition, when it is divided into X = X1 · 2 ³² + X2, Z = P · 2 ³²
+ (R · 2 ³² + X2) / Y. Here, if the first bit of R is “1”, the calculation result of (R · 2 ³² + X2) / Y becomes a negative value, resulting in an erroneous result. is not.

(B) Case of d (Y) ≧ 32 This is a case where the divisor Y, which is a problem of the above (ii), cannot be represented by 4 bytes.

 Therefore, the following U and V are considered.

U = [X / 2 ^{d (Y) -31} ] V = [Y / 2 ^{d (Y) -31} ].

Since U ≦ X / 2 ^{d (Y) −31} <U + 1, V ≦ Y / 2 ^{d (Y) −31} <V + 1, Assuming that U = P · V + R (0 ≦ R <V), Here, since R <V, Therefore, the above equation is Becomes

1) When P ≦ R Z = [X / Y] = P 2) When P> R Here, from ^{2 d (Y) -1 ≦ Y} <2 d (Y), 2 30 ≦ V = [Y / 2 d (Y) -31] <2 31 2 60 ≦ V 2 <2 62 Meanwhile, U = [X / ^{2d (Y) -31} ] = [X / ^{2d (X) -d-31} ] = [X · ^{2d + 31} / ^{2d (x)} ], ^{2d + 30} ≦ U < 2 ^{d + 31} Therefore, from U / V ² <2 ^{d + 31/260} = 2 ^d-29 or more, P−2 ^d−29 <P− (P−R) / (X + 1) <X / Y d = d (X) −d (Y) ≦ 32 (∵d (X) ≦ 64, d (Y)
≧ 32) From d ≦ 29, if P−1 ≦ Z ≦ Pd = 30, if P−2 ≦ Z ≦ Pd = 31, if P−4 ≦ Z ≦ Pd = 32, then P−8 ≦ Z ≦ P That is, (1) When d ≦ 29 If X <P · Y, then Z = P−1 If X ≧ P · Y, then X = P (2) If d = 30 X If <(P-1) .Y, then Z = P-2 (P-1) .Y≤X <P.Y, if Z = P-1 P.Y ≤X, then Z = P (3 ) When d = 31 If X <(P−3) · Y, then Z = P−4 (P−3) · Y ≦ X <(P−2) · Y If Z = P−3 (P -2) · Y ≦ X <(P-1) · Y, then Z = P-2 (P-1) · Y ≦ X <P · Y, Z = P-1 if P · Y ≦ X If, in this case of Z = P (4) d = 32, a case of X = -2 ^63, is already considered already.

Utilizing the above-described operation principle, 63-bit integer type division is performed as in the processing flow shown in FIG. Hereinafter, the fourth
The processing will be described according to the processing shown in FIG.

The number of significant digits of the dividend X is entered in the variable DX, and the number of significant digits of the divisor Y is entered in the variable DY. Put the difference between DX and DY in variable D.

When DY = 0, a division error occurs.

A trivial quotient Z according to the number of significant digits DX and DY and the difference D
Ask for.

For X = -2 ^63, i.e., the case of X = -2147483648, putting the quotient of (X / 2) / Y to P, put the remainder of (X / 2) / Y in R.

 If R * 2 <Y, Z = P * 2. If R * 2 ≧ Y, Z = P * 2 + 1.

When DY ≦ 31, the lower 32 bits of Y are put in Y2 and the following processing is performed.

If D ≦ 31, Z = X / Y2.

• Put the upper 33 bits of X into X1.

・ Put the lower 31 bits of X into X2.

・ Put X1 / Y2 in Z.

Shift Z 31 bits to the left.

• Add X2 / Y2 to Z to obtain the desired quotient.

If DY> 31, the following processing is performed.

• Put X's right (DY-31) bit shifted into U's.

• Put the result of shifting Y to the right (DY-31) bits into V.

・ Enter the quotient of U / V in Z. This is P.

・ Put X-P * Y in R.

Next, while R <0, the process of subtracting 1 from Z and adding Y to R is repeated.

 If R becomes 0 or a positive number, the value of Z at that time is the quotient to be obtained.

〔The invention's effect〕

As described above, according to the present invention, for example, one word is
On 32-bit computers, 63-bit integer division can be performed without using floating-point instructions.
This makes it possible to relax specification restrictions in languages and the like. Therefore, the calculation accuracy of the application program can be improved, and the processing load can be reduced.

[Brief description of the drawings]

Fig. 1 is an example of the configuration of the present invention. Fig. 2 is an example of an application system according to an embodiment of the present invention. Fig. 3 is an example of processing according to an embodiment of the present invention. Fig. 4 is an embodiment of the present invention. 4 shows an example processing flow. In the figure, 10 is a computer, P1 to P3 are processing steps, X is a dividend, and Y is a divisor.

Claims (1)

(57) [Claims] 1. Calculation of a quotient and a remainder of a division of a binary integer in which a dividend and a divisor are each represented in a range of 2n bits using a computer having a division instruction execution mechanism for dividing by an n-bit binary integer. A binary integer division processing method in which processing is performed by software, the process of obtaining the number of significant digits of a binary integer of a dividend X and a divisor Y, and the difference between the number of significant digits of the dividend X and the divisor Y and the number of significant digits. Divides the case according to the process of finding the answer of the trivial division and whether the number of significant digits of the divisor Y is less than or equal to n-1 or greater than or equal to n. The answer of the division is obtained by the above-mentioned division instruction execution mechanism. If the answer is not less than n, the dividend X and the divisor Y are shifted to obtain values each having the number of significant digits of n bits or less. Is divided by the above division instruction execution mechanism using As long as the remainder, which is the value obtained by subtracting the product of the quotient and the divisor Y obtained from the number X, is negative, the obtained quotient is reduced by 1 and the divisor Y is added to the remainder, thereby repeating the above. A binary integer division processing method, comprising: a process of obtaining a quotient and a remainder of division.