JPH0419571B2 - - Google Patents

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 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-sized computers because it can share logic with multiplication and can increase speed. Simply put, the principle is to treat the dividend and divisor as the numerator and denominator of a small number, multiply the numerator and denominator by the same convergence coefficient until the denominator of the fraction approaches 1, and then use the final numerator as the desired quotient. It is something. This method is used, for example, by Kai Wang.
Wang), translated by Yasushi Horikoshi et al., “High-speed calculation method for computers”, Modern Scientist (September 1, 1980), p. 251
~254. The conventional convergent division method shown in the above-mentioned literature will be explained using the calculation block diagram shown in FIG. Here, the positive numbers N ₀ and D ₀ are normalized
Let us calculate the divisor value Q _x with and as the dividend and divisor, respectively. N ₀ and D ₀ are, for example, 0.5≦1 in binary representation, and N ₀ <0.5≦D ₀ <1. Q _x , N ₀ ,
The relationship of D ₀ is expressed by equation (1). Q _x = N ₀ /D ₀ ...(1) First, the initial value R ₀ of the convergence coefficient is found from the following equation (2) (step). R ₀ =1−D ₀ (2) Next, the dividend N ₀ and the divisor D ₀ are updated for the first time using the initial value R ₀ (step). Here, the basic formulas for updating the dividend and divisor are respectively
This is shown in equations (3) and (4). Note that N _i is the dividend of the i-th update, D _i is the divisor of the i-th update, R _i is the convergence coefficient of the i-th update, and i=0, 1, . . . . N _i・R _i +N _i →N _i+1 ...(3) D _i・R _i +D _i →D _i+1 ...(4) Since i=0 now, the dividend of the first update
N ₁ and divisor D ₁ are obtained based on equations (3) and (4). In other words, from N ₀・R ₀ +N ₀ , N ₁ becomes D ₀・R ₀
D ₁ is obtained from +D ₀ by a product-sum operation. Next, the convergence coefficient is updated for the first time (step). Here, the basic formula for updating the convergence coefficient is
It is shown in equation (5). 1−D _i+1 →R _i+1 ...(5) Once the dividend N ₁ , divisor D ₁ , and convergence coefficient R ₁ for the first update are determined, using these values, (3), (4 ),
Based on each formula (5), the dividend number N ₂ of the second update,
Find the divisor D ₂ and the convergence coefficient R ₂ (step →
→→→). Then, such updating calculations of the dividend and divisor and updating calculations of the convergence coefficient are continued until the updated value D _i+1 of the divisor becomes D _i+1 within the precision of the multiplication device used for the product-sum calculation (step → →→
→ routine), when N _i+1 →1 [(1)
The division was completed by using the updated value of the numerator of the equation 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 D _i+1 of the divisor is equal to the constant 1, the accuracy of the arithmetic unit is is finite, so the determination was made based on D _i+1 =1+ΔQ (ΔQ is the determination error). However, the determination error ΔQ differs depending on the input data, and there is a problem that 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,
It is an object of the present invention to provide a convergent division method that performs division by the minimum number of calculation steps determined from the input data width. (Means for Solving the Problems) The present invention includes a multiplication circuit that multiplies floating point data and an arithmetic operation circuit that adds and subtracts floating point data. This relates to a division method in a division device that performs division using B ₀ ) and dividend (A ₀ ), and in order to solve the problems of the prior art,
It was constructed to include the following four steps. In the first step, the exponent part of the divisor (B ₀ ) is calculated from the exponent part of the dividend (A ₀ ) to obtain the dividend (A ₀ ') in which only the exponent part of the dividend (A ₀ ) is updated. Find the divisor (B _{0 '} ) by resetting and updating only the exponent part of the divisor (B ₀ ). In the second step, the updated dividend (A ₀ ″) is obtained by multiplying the dividend (A 0 ′) with only the exponent part updated by a constant, and the updated dividend (A ₀ ′) with only the exponent part updated is multiplied by a _constant . After multiplying, the value obtained by subtracting from another constant (1−4/3B ₀ ′) is set as the initial value of the convergence coefficient (X ₀ ).In the third step, the divisor (B ₀ ) or Number of update operations performed in the next fourth step based on the data word length of the mantissa of the dividend (A ₀ )
Find (m). In the fourth step, m updating operations are sequentially performed on the dividend (A ₀ ″) updated in the second step, and the updated dividend (A _m
=A _m-1・S _m-1 +A _m-1 ) is the quotient. Of the m update operations performed in this fourth step, the first update operation is to combine the dividend (A ₀ ″) updated in the second step with the initial value of the convergence coefficient (S ₀ ). Product by multiplication (A ₀ ″S ₀ ) and the updated dividend
(A ₀ ″) and the sum (A ₀ ″S ₀ +A ₀ ″) is the first update dividend (A ₁ ). From the second time onwards, for example, in the i-th update operation, Update dividend obtained from the previous update operation
⁽ _{A i - 1 _}
S _i-1 ) and the updated dividend (A _i-1 ) obtained from the previous update calculation (A _i-1・S _i-1 +A _i-1 )
is determined as the new updated dividend (A _i ). (Function) Since the processing of the exponent part is completed by the processing of the first step, the calculation processing is thereafter performed using floating point data, but conventional fixed point processing can be applied to the mantissa processing. In the second step, variables are converted so that the mantissa value range of the divisor is 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, and this number can be determined from the divisor or the mantissa data word length of the dividend.
Unlike the conventional method, it is no longer necessary to perform divisor update calculations 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 updating 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 reduced. (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, it is assumed that the division value Q _F is calculated using the data A ₀ and B ₀ that have been normalized in floating point representation as the dividend and the divisor, respectively. However, B ₀ >0. The relationship among A ₀ , B ₀ , and Q _F is expressed by equation (6). Q _F =A ₀ /B ₀ (6) The principle of the division operation in this embodiment is based on a 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 dividend
Calculate the exponent part of the divisor B ₀ from the exponent part of A ₀ to obtain an updated value A ₀ ' that updates only the exponent part of the dividend A ₀ ,
After the subtraction, an updated value B ₀ ' is obtained by resetting only the exponent part of the divisor B ₀ . After that, the range of the updated divisor and the mantissa part of the dividend is converted using a linear equation (A ₀ ″, B ₀ ″), and the divisor B ₀ ″ that has undergone the conversion process is Multiply the dividend A ₀ ″ sequentially as a coefficient to multiply the reciprocal asymptotic value,
The dividend update calculation 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 set such that when the divisor B ₀ '' is successively multiplied by the coefficient to be multiplied by the above number of times, the multiplication result becomes 1.The minimum necessary for the above update operation. The method for determining the number of times will be described below. First, consider applying the following transformation formula (formula (7)) using infinite products to the calculation of formula (6).

[Formula] Applying this equation (7) directly to equation (6), Q _F =A ₀ /B ₀ =A ₀ ′/B ₀ ′=A ₀ ′/1−X =A ₀ ′(1+X)( ^{1 + _} _{_ _ _}
Since the value is the same as the mantissa of , 0.5≦B ₀ ′<1 (9) and the range of variable It becomes a state where sex is not good. For this reason, the variable
Let us consider converting B ₀ ' to a variable S having a zero-point symmetric range by conversion using a linear equation. If we set S=1−4/3B ₀ ′ (11), the range of B ₀ ′ is 0.5B ₀ ′<1, so −1/3<S1/3, and if we use this S, (6 ) formula is Q _F =A ₀ /B ₀ =A ₀ '/B ₀ ' = 3/4A ₀ '(1+S)(1+S ² )(1+S ⁴ )...(1+S ²ⁿ )... (12) The conversion formula of equation (7) can be applied with the fastest convergence speed. On the other hand, as shown ⁱⁿ Figure 2, the mantissa data word length is a finite value of M ^bits , so the expression
If we multiply the parentheses in (12) a finite number of times, we get M ^bit
This is the limit value that can be expressed as Therefore, we sequentially expand the infinite multiplication product of equation (12) and calculate the multiplication value up to the mth time by Q _m
Then, equation (12) can be approximated by the following finite series. This is the limit value that can be expressed in M ^bits . Therefore, formula (12)
Sequentially expand the infinite product of and calculate the multiplication value up to the mth
Assuming Q _m , equation (12) can be approximated by the following finite series.

[Formula] Therefore, if ΔQ _{m, which is the difference between the infinite product of formula (12) and the finite series of formula (13), is ΔQ m =} |Q _F −Q _m | (14), then formula (14) What is necessary is to find out how many times the maximum value ΔQ _m (MAX) of ΔQ m (MAX) falls outside the range of M ^bits . ΔQ _m (MAX) is the mantissa part of |A ₀ ′ |→1, S→
1/3, and on the other hand, if the numerical limit of a binary number that can be expressed with a width of M ^bits is 2 ^-p , then ΔQ _m (MAX) is

[Equation] (15) Therefore, p for M ^bit word length accuracy is p≧(M+1)(log ₃ 2) + 1 (16), and t in Equation (13) is the lower limit of p ( M+2)
The number of terms m that is equal to (log ₃ 2) + 1 is the value to be sought. Therefore, from 2 ^m −1=(M+2)(log ₃ 2)+1 (17), m=log ₂ {(M+2)(log ₃ 2)+2} (18), and for example, the mantissa is S ^bit ( In the data format of sign) + 15 ^bits (data), m = 3.7≒4 from M = 15.
It becomes a step. Also, since equation (12) is a convergent series, S is 0
If it is close to , it converges by multiplying the convergence coefficient m or less times, but even after convergence, even if the coefficient is multiplied, it is outside the range of M ^bits , so Q _m of the operation result does not diverge. Therefore, the number m of update operations of multiplication and addition for updating the dividend can be determined from the mantissa data word length M ^bits of the input data. Next, the specific procedure of the division operation according to this embodiment will be described based on FIG. First, process the exponent part as described above,
Dividend A _{0 ′} where only the exponent part of dividend A ₀ is updated,
Find the divisor B ₀ ' in which only the exponent part of the divisor B ₀ is updated (reset). ). Next, using the updated divisor B ₀ ', the initial value S ₀ of the convergence coefficient is determined from the following equation (14) (step). 1-4/3B ₀ ′= ⁰ (19) Furthermore, the updated dividend A ₀ ′ is multiplied by a constant (4/3) to obtain a dividend A ₀ ″ whose range has been converted (step). Using this S ₀ , update the updated dividend A ₀ ″ as follows:
This is performed by the product-sum operation (multiplication and addition) of equation (20) (step). A ₀ ″S ₀ +A ₀ ″=A ₁ (20) Next, the convergence coefficient S ₀ is updated by multiplying the equation (21) (step). S ₀ ² =S ₁ (21) Hereafter, the updating calculation of the dividend and the updating calculation of the convergence coefficient are performed m times (step→→). If the value of m obtained by the method described above is used, the calculation result A of the final time, that is, the m-th product-sum operation A _m-1・S _m +A _m-1 = A _m (22) The division is completed when _m becomes the desired quotient. In the example described above, as the condition of equation (6),
The divisor B ₀ is limited to positive, but if the divisor B ₀ is negative, the polarity of both the divisor and dividend may be reversed, and then the same calculation process as above may be performed.
No problem. Furthermore, after the division is completed, the remainder is handled using the following formula:
It can be obtained by performing multiplication and subtraction operations based on (23). Dividend A ₀ − Divisor B ₀ × Quotient = Remainder (23) (Effects of the Invention) According to the present invention, the minimum necessary update is performed from the viewpoint of calculation accuracy of the data word length and finite word length of the mantissa part of the divisor or dividend. Since I decided to calculate the number of operations,
The divisor update operation required in the conventional convergent division method can be omitted, and the advantage is that the number of calculation steps and calculation time are significantly reduced, making it possible to speed up the division process. be.
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 the drawing]

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

Claims (1)

[Scope of Claims] 1. A multiplication circuit that performs multiplication of floating-point data and an arithmetic operation circuit that performs addition and subtraction of floating-point data, and performs division using a divisor and a dividend that are represented as floating-point numbers. In the division method of the division device, the exponent part of the divisor is subtracted from the exponent part of the dividend to obtain a dividend with only the exponent part of the dividend updated, and the divisor with only the exponent part of the divisor updated is obtained. 1 step, multiply the dividend whose exponent part only is updated by the constant 4/3 to obtain the updated dividend, reset only the exponent part, multiply the updated divisor by the constant 4/3, and then multiply from the constant 1. a second step in which the subtracted value is used as the convergence coefficient and initial value; and a number of update operations to be performed on the dividend updated in the second step based on the mantissa data word length of the divisor or dividend. The third step to obtain The sum of the dividend updated in step is the first
The second and subsequent update calculations are performed by multiplying the update dividend obtained from the previous update calculation by the update convergence coefficient obtained by squaring the previous update convergence coefficient and the previous update dividend. This is done by adding the sum of the updated dividend obtained from the updating calculation as a new updated dividend, and the above updating calculation is performed sequentially the number of times obtained in the third step, and the finally obtained updated dividend is A division method characterized by comprising a fourth step in which the quotient is a quotient.