JPS6286434A - Division system - Google Patents

Abstract

PURPOSE:To reduce an operating time by applying updating operation sequentially by the number of times obtained by the 3rd step sequentially and using an updated dividend obtained finally as a quotient to reduce the number of arithmetic steps. CONSTITUTION:In the 3rd step, based on the data word length of a mantissa part of a divider (B0) or a dividend (A0), the number of times (m) of the updating operation executed in the next 4th step. In the 4th step, m-time of updating operation is applied sequentially to the dividend (A0'') revised by the 2nd step and the updated dividend (Am=Am-1.Sm-1+Am-1) obtained finally is used as a quotient. Among the m-time updating operation executed in the 4th step, the 1st updating operation a product (A0''S0) obtained by multiplying the (A0'') updated at the 2nd step by the initial value (S0) of the convergent coefficient, and the result is added with the dividend (A0''), then the sum (A0''S0+A0'') is obtained, then it is used as the 1st updated dividend. Thus, the division processing is quickened by the remarkable reduction in the number of arithmetic steps and the further reduction of the operating time.

Description

Detailed Description of the Invention (Industrial Application Field) The present invention relates to a division method, and more particularly, to a convergent division method in a division device equipped with a floating-point arithmetic multiplication circuit and an η-arithmetic operation circuit. .

(Prior Art) There is a convergent division method as a division method by a computer. This method is classified as a multiplicative division method, and is also used in large-scale computers because it can share the logic with multiplication and can increase speed. Simply stated, the dividend and divisor are considered to be the numerator and denominator of a decimal number, and the numerator and denominator are multiplied by the same convergence coefficient by 1, where the denominator of the fraction approaches 1, and the final numerator is the desired quotient. It is something to do.

This method is shown, for example, in ``High-speed Computing Method for Computers'' by Kaj Wang, translated by Yasushi Horikoshi et al., Kindai Kagakusha (September 1, 1995), p. 251-254K.

The conventional convergent division method shown in the above document will be explained using the calculation block diagram shown in FIG. Here, a positive number N that is being normalized. bird.

Let us calculate the divisor value Qx with and as the dividend and divisor, respectively. For example, No and Do are 0.5 in binary representation.
≦No<1. o5≦Doku1. QX +No, D
The relationship between o is expressed by equation (1).

Qx = No/Do.(1) First, the initial value R6 of the convergence coefficient is obtained from the following equation (2) (step Φp).

R=1-D.

O...(2) Next, the dividend N is calculated using the initial value R6. and the divisor. the first of
The second update is performed (steps ■, O). Here, the basic formulas for updating the dividend and divisor are equations (3) and (4), respectively.
) is shown in the formula. Note that N is the dividend of the first update, and D
Let i be the divisor of the i-th update, Ri multiplied by the convergence coefficient of the i-th update, and i=o, I, .

NH-R40 N1 →N1+1
...(3) Di"Rj +Df→DN+
(4) Since i = O now, the dividend N1 and divisor D1 of the first update are found based on equations (3) and (4). That is, N. -R
o ten N. N1 is obtained from Do-Ro+Do, and Dl is obtained by sum-of-products calculation.

Next, perform the first update of the convergence coefficient (step [
phase]). Here, the basic equation for updating the convergence coefficient is shown in equation (5).

1 − I)j++ →Ri+i
°−(5) First update dividend N1. Divisor D1. Once the convergence coefficient R1 is determined, using these values, the second update dividend N2 + divisor D2. Find the convergence coefficient R2 (step ■ → [phase] → O → [phase]
→■). Then, such updating calculations of the dividend and divisor and updating calculations of the convergence coefficient are performed until the updated value Di+1 of the divisor becomes Di+1 → 1 within the precision of the multiplication device used for the product-sum calculation, that is, [the denominator of equation (1)] Update value] → Continue until it becomes 1 (step @ → @ → O- + [phase] → ■ routine),
The division is completed by using [the updated value of the numerator in equation (1)] when Di-t-1→1 as the quotient (step @).

(Problem to be solved by the invention) However, in the conventional convergent division method described above,
When determining whether the updated value Di+1 of the divisor is equal to the constant 1, since the accuracy of the arithmetic device is finite, Di
Judgment was made based on +1 "" 1+ΔQ (ΔQ is the judgment error). However, there are problems in that the judgment error ΔQ varies depending on the input data, and the number of update operations i is not constant. Furthermore, while the final quotient corresponds to the updated value of the dividend, there are three types of update operations: the dividend, the divisor, and the convergence coefficient, which has the drawback of increasing the number of calculation steps and the calculation time. Ta.

Therefore, the present invention eliminates the problem that the number of operations is not constant and the disadvantage that the number of operation steps is large in the conventional convergent division method described above, and divides by the minimum number of operation steps determined from the input data width. The purpose of this paper is to provide a convergent division method that performs the following.

(Means for Solving the Problem) The present invention includes a multiplication circuit that multiplies floating point data and an arithmetic operation circuit that adds and subtracts floating point data. BO)
This invention relates to a division method in a division device that performs division using a dividend (AO) and a dividend (AO), and is configured to include the following four steps in order to solve the problems of the prior art.

In the first step, the exponent part of the divisor (BO) is subtracted from the exponent part of the dividend (AO) to obtain the dividend (Ao') in which only the exponent part of the dividend (AO) has been updated. The divisor (BO
').

In the second step, the dividend (
AO') K constant (for example, 4/3) is multiplied to obtain the updated dividend (A, 1/), and the divisor (A, 1/) with only the exponent part updated is
go') Multiplyed by a VC constant (for example 4/3) and then subtracted from another constant (for example 1) (1,
--HB. ') is the initial value (XO) of the convergence coefficient.

In the third step, the divisor (BO) is still the dividend (AO
), the number of update operations (m) to be executed in the next fourth step is determined based on the data word length of the mantissa part.

In the fourth step, m update operations are sequentially performed on the dividend (Ao/l) updated in the second step, and the updated dividend (Am=Arn-4') is finally obtained.
5m-1+Am-+) is the quotient. Of the m update operations performed in the fourth step, the first update operation is the dividend (AO”) updated in the second step.
) by the initial value of the convergence coefficient (So) (Ao
``So)'' and the updated dividend (AO'') (
Ao″So+Ao″) is the first updated dividend (A1)
shall be.

From the second time onwards, for example, in the first update calculation, the convergence coefficient (5i
-1=si-2) (A1-+・Sl-
+ ) and the updated dividend (A
i-1) and the sum (Ai-+・5i-10A4-
+) as the new updated dividend (At).

(Operation) Since the processing of the exponent part is completed by the processing of the first step, conventional fixed-point processing can be applied to the mantissa processing from now on, although the calculation processing is performed using floating-point data.

In the second step, variable conversion is performed so that the mantissa ranges of the dividend and divisor are symmetrical about the origin, and the number of convergence operations can be kept to the minimum necessary.

In the third step, the number of update operations that are performed sequentially on the dividend updated in the second step is determined, but since this number can be determined from the divisor or the mantissa data word length of the dividend, the conventional method As shown in FIG. 2, it becomes unnecessary to perform the divisor update operation one after another and determine whether it is equal to the constant 1 each time, and the procedure for determining the divisor update value itself can be omitted.

In the fourth step, the number of calculation steps can be reduced by simply performing the update operation on the dividend updated in the second step the number of times determined in the third step to obtain the quotient to be determined. At the same time, calculation time can be shortened.

(Embodiment) A division method according to an embodiment of the present invention will be described below with reference to FIGS. 1 and 2. FIG. 1 is a block diagram showing the basic calculation contents, and FIG. 2 is a diagram showing the data format of floating point representation used in the calculation of this embodiment.

Here, we will calculate the division value Q2 using r (1 data A. and B. as the dividend and divisor, respectively, which are normalized in floating point representation. However, it is assumed that B.>0.AO
The relationship between + Bo + QIN is expressed by equation (6).

QF −Ao / Bo −(
6) The principle of the division operation in this embodiment is based on convergent algorithm, and first an overview of the basic operation contents will be described. In the division operation of this embodiment, the exponent part is processed first. That is, the divisor B is obtained from the exponent part of the dividend Ao. Dividend A is obtained by subtracting the exponent part of . An updated value A in which only the exponent part of is updated. ′ is obtained, and after the subtraction, the divisor B is obtained. Updated value B that resets and updates only the exponent part of . Find ′. after that,
Range conversion for the mantissa parts of the updated divisor and dividend is performed using a linear expression (Ao//, Bo//), thereby speeding up the number of convergence operations in subsequent convergence operations. Divisor B that receives the conversion process. Dividend A as a coefficient to be multiplied by the reciprocal asymptotic value of '. ′
is successively multiplied by , the dividend update operation is repeated a predetermined number of times, and the final updated value of the dividend is taken as the quotient. Here, the number of update operations to be executed is the divisor B. The setting is such that when two sets of coefficients are sequentially multiplied by '', the multiplication result becomes 1.

” How to determine the minimum number of two-set update operations is described below. First, consider applying the following transformation formula (formula (7)) using an infinite product to the calculation of formula (6).

If we apply this equation (7) directly to equation (6), we get the divisor B. The mantissa of Then, 0.5 < b
. < 1 (9), so the range of the variable In this case, consider converting the variable B.' into a variable S having a range symmetric at the zero point by conversion using a linear equation.

If we set S=1-, Bo', then the range of Bo' is 0.5 < B. ′<1, so if S is used, equation (6) becomes, and the conversion formula of equation (7) can be applied with the fastest convergence speed.

On the other hand, the mantissa data word length is Mb
Since it is a finite value, it becomes a limit value that can be expressed as J+It by multiplying the parentheses of 20 by a finite number of times with a width of Mbit. Therefore, if we sequentially expand the infinite product of formula 0■ and let Qm be the multiplication value up to the mth time, 2α→ becomes
It can be approximated by the following finite series.

Therefore, ΔQm, which is the difference between the infinite product of formula 0 and the finite series of 203, is calculated as ΔQm−IQF Qm l (J
4, it is only necessary to find out how many times the maximum value ΔQm(MAX) of equation (1Φ) is multiplied to be outside the range of Mbit.

ΔQm(MAX) is lAo'l→1. This occurs when X → 1/3, and on the other hand, if the numerical limit of the binary number that can be expressed by the width of Ml)It is 2"-111, then ΔQm(MAX) is
Therefore, p for the word length precision of Mb i t is p
≧(Ml2) (log32)+1 04
Then, t in equation (13) is the lower limit value (Ml 2X/-o
g32) → The number of terms m that is equal to −1 becomes the value to be sought.

Therefore, 2′1″−1,=(Ml2) (Aog32)
+1 From α, m = Aog2((Ml2)(log32)+2)
(IQ, for example, the mantissa is skl+
In the data format of t (sign) + 151) It (data), it takes m-3°7≠4 steps since M=15.

In addition, since the formula 0→ is a convergence series, if The number of operations m can be determined from the mantissa data word length Mbtht of the input data.

Next, the specific procedure of the division operation according to this embodiment will be described based on FIG.

First, the exponent part is processed as described above, and the dividend A is
. Dividend A with only the exponent part updated. ′ and Hi divisor B. Update only the exponent part of Bo
′ 4 (Step ■).

Next, update the divisor B. ′, the initial value of the convergence coefficient
. is obtained from the following (1◆formula) (step ■).

Dividend A whose range has been converted by multiplying i. ′ (
Step ■).

The S. Update dividend A using . ′ update as follows (a)
This is performed by the product-sum operation (multiplication and addition) of the expression (step ■).

AO"SO1A.-A1e;!fJ) Next, the convergence coefficient S. is updated by multiplying the equation QI) (step 2).

5o-81e1) Under IJ, the update calculation of the dividend and the update calculation of the convergence coefficient are performed m times (steps ■→■→■). The value of this m is
Use what you have in a two-pronged way. Then, the final time, that is, the m-th product-sum operation Am-1 sm I Am-1
= Am The calculation result Am of eA becomes the desired quotient and the division is completed.

2. In the example mentioned above, as a condition of equation (6), the divisor B
o is positively limited, but the divisor B. If is negative, the polarity of both the divisor and the dividend can be inverted, and then the same arithmetic processing as described above can be performed, and no problem will arise.

Furthermore, the remainder may be handled by performing a multiplication operation based on the following equation (a) after the division is completed.

Dividend A. -Divisor B. × Quotient - Remainder (c) (Effects of the Invention) According to the present invention, the minimum number of update operations required is determined from the data word length of the mantissa of the divisor or dividend and the calculation accuracy of the finite word length. As a result, the divisor update operation required in conventional convergent division methods can be omitted, and the number of calculation steps and calculation time are further reduced, making it possible to speed up the division process. There are some advantages.

In addition, in the division device to which the division method of the present invention is applied, division can be performed simply by using a multiplication circuit and an arithmetic operation circuit, so there is no need to provide a special calculation section dedicated to division, and the device can be made smaller and more economical. There are advantages.

[Brief explanation of drawings]

Fig. 1 is a block diagram showing the main calculation contents of the embodiment of the present invention, Fig. 2 is a diagram showing the data format of floating point representation used in the calculation of the above embodiment, and Fig. 3 is a diagram of the conventional calculation method. It is a block diagram showing the contents.

Claims (1)

[Scope of Claims] A division operation that performs division using a divisor and a dividend expressed in floating point numbers, comprising a multiplication circuit that multiplies data expressed in floating point numbers, and an arithmetic operation circuit that performs addition and subtraction of data expressed in floating point numbers. In the division method in the device, the exponent part of the divisor is subtracted from the exponent part of the dividend to obtain a dividend in which only the exponent part of the dividend is updated, and the first step is to obtain a divisor in which only the exponent part of the divisor is updated. step, multiplying the dividend with only the exponent part updated by a constant to obtain an updated dividend, resetting only the exponent part, multiplying the updated divisor by a constant, and then subtracting it from another constant. a second step of setting the convergence coefficient as an initial value, and based on the mantissa data word length of the divisor or dividend,
A third step of calculating the number of update operations to be performed on the dividend updated in the second step, and the first update operation is performed using the dividend updated in the second step and This is done by calculating the sum of the product multiplied by the initial value of the convergence coefficient and the dividend updated in the second step as the first updated dividend. The sum of the product obtained by multiplying the updated dividend obtained from the previous update calculation by the update convergence coefficient obtained by squaring the convergence coefficient of the previous time, and the updated dividend obtained from the previous update calculation is calculated as the new updated dividend. and performing the update operation the number of times determined in the third step,
A division method characterized by comprising: a fourth step in which the updated dividend finally obtained is used as a quotient;