JP2546916B2 - Trigonometric function operation unit - Google Patents

Description

The present invention relates to a trigonometric function operation device.

[Conventional technology]

Conventionally, as a method for calculating the trigonometric function, the following equation Taylor series expansion ^{That, sinθ = θ-θ 3/} 3! + Θ 5/5! -Θ 7/7! + Or using ...... a (1), or , Continuous fraction expansion, Chebyshev series expansion, etc. are used to obtain rational function approximation.

However, in any of the above methods, there is a problem that the calculation time becomes long because many multiplications or divisions are required, and the calculation speed is not satisfied.

As a measure against the above problem, as a trigonometric function operation method suitable for a microprogram-controlled computer,
There is a CORDIC (Cordinate Rotation Digita Computer) algorithm. It is known that this CORTIC algorithm is effective even in a computer that does not have a high-speed multiplier because it can perform operations by addition, subtraction, and right shift.

The principle of subtraction by the CORDIC algorithm is as follows.

In the binary system, n (positive integer) digit precision sin θ and c
When obtaining osθ, the angle θ is expressed by the following equation using the constant γ _k and the sequence {a _k }.

θ = a ₀ × γ ₀ + a ₁ × γ ₁ −a ₂ × γ ₂ + ... + a _n-1 × γ _n-1 + ε (2) where γ _k = arctan (2 ^-k ) ...... ( 3) _ak = {+1, -1} (4) The sequence {a _k } is a method similar to the division in the separation method and can be uniquely determined. The process of determining this sequence {a _k } is called pseudo division.

Therefore, sin θ and cos θ can be obtained as follows by the processing theorem when R _k = 1 / (1 + 2 ^−2k ) ^{1 ] 2} (5).

When a _k = + 1, φ _{k + 1} = φ _k + γ _k cos (φ _{k + 1} ) = R _k (cos φ _k −2 ^−k × sin φ _k ) ... (6) sin (φ _{k + 1} ) ^{_{= R k (sinφ k -2 -k}} × cosφ k) ...... (7) a k = -1 when _{the, φ k + 1 = φ k} -γ k cos (φ k + 1) = R k (cosφ ^{_{k -2 -k × sinφ k) ......}} (8) sin (φ k + 1) = R k (sinφ k -2 -k × cosφ k) a ... (9). Here, with R ₀ as an initial value, φ _k is brought closer to θ by repeatedly executing the above arithmetic operation so as to increase the value of _k . In this way, sin θ and cos θ are obtained through the operation. This calculation process is called pseudo multiplication.

The following is a description of the floating-point operation principle of trigonometric functions using the CORDIC algorithm, using mathematical expressions.

[1] The angle that is the input data is θ (0 <θ <π / 2).

[2] Assuming that the angle θ is θ = 2 ⁻ⁱ × Θ (1 ≦ Θ <2, i is an integer), i (exponent part) and Θ (mantissa part) are determined (floating point),
Let V _i = Θ be the initial value.

[3] The following [4] is repeated for k = i, i + 1, i + 2, ..., M-1 (m is a positive integer).

[4] If V _k ≧ 0, then if a _k = + 1 V _k <0, then set a _k = −1 and V _{k + 1} = 2 × (V _k −a _k × Γ _k ) ... (10) However, Γ _k = 2 ^k-1 × arctan (2 ^-k ) ... (11) [5] X _m = V _m × (1 / K), Y _m = V _m …… (12) are initial values. To do. here, Is a constant that satisfies.

[6] For k = m, m-1, m-2, ..., i + 1,
Repeat [7] below.

[7] X _k-1 = X _K −a _k × 2 ^−2k × Y _k (14) Y _k-1 = (Y _k + a _k × X _k ) / 2 (15) [8] sin θ = 2 ⁱ × Y _i and cos θ = X _i are obtained at the same time.

When the above CORDIC algorithm is used in a computer, the constant of the above equation (11) Γ _k = 2 ^k-1 × arctan (2 ^−k ) (k = 0,1, ..., m−1) and the above (13) A constant of 1 / K, which is the reciprocal of K = Π ^m (1 + 2 ^-2k ) ^{1 ] 2} (k = 0,1, ..., n), is required.

On the other hand, when this algorithm is used, the range of θ is applicable only when 0 <θ <π / 4. Therefore, when π / 4 or more, the remainder of the input data is taken at π / 4, and This algorithm will be used for values. Therefore, the data input to this algorithm has an exponent of −
The range is from m to -1.

This is clear from the fact that π / 4 = 3.14 …… / 4 ＝ 0.785 …… ＝ 1.57 …… × 2 ⁻¹ .

As described above, when the exponent part changes from -m to -1, m constant values are also required. That is, the above (11)
The constant Γ _{k of the} equation and the constant 1 / K obtained from the equation (13) are
You must prepare m pieces each. In a conventional computer, the above constants are stored in advance in a constant ROM (Read Only Memol).
Prepared in y) and used for the calculation processing of trigonometric functions. An example of the contents of the constant ROM of Γ _k in this case is shown in the table of FIG. 3, and an example of the contents of the constant ROM of the constant 1 / K is shown in FIG. However, it is a table showing constants when the mantissa calculation range is from −32 to −1 (decimal number),
It is shown in hexadecimal notation.

A conventional trigonometric function operation device using the CORDIC algorithm is configured as shown in FIG. 2 as an example.

As shown in FIG. 2, this conventional example is based on an exponent calculator.
14, constant ROMs 15 and 16, adder / subtractors 17 and 20, registers 18 and 21, barrel register 19, stack 22 and multiplier 23. The operation will be described below with reference to FIG.

A value of θ = 2 ⁻ⁱ × Θ (1 ≦ Θ <2, i is an integer) expressed by a binary floating point number is input from the bus 103, and the exponent part i is set to the exponent part calculator 1
Enter in 4. The mantissa part Θ of θ is input to the register 21.

The following operation is repeatedly executed while incrementing the output value k of the exponent calculator 14 to k = i, i + 1, i + 2, ..., M-1.

The V _k value stored in the register 21 is transferred to the input A of the adder / subtractor 20 via the bus 105, and at the same time, the value in the constant ROM 16 of the constant Γ _k selected by the address obtained by the exponent calculator 14 Is transferred to B of the adder / subtractor 20 via the bus 104. At the same time, the operation of the adder / subtractor 20 is controlled by the sign of the sign digit of the register 21, and the output is shifted to the left by one digit and written in the register 21.

The V _m value is fetched from the register 21 via the bus 104 and transferred to the register 18 and the multiplier 23. Constant ROM of constant 1 / K selected by the address calculated by exponent calculator 14
Transfer the value at 15 to the multiplier 23 via bus 104,
The product V _m / K of V _m and 1 / K obtained in the multiplier 23 is transferred to the register 21 via the bus 104.

 The following operation is repeatedly executed while decrementing the output value k of the exponent calculator 14 as k = m, m-1, m-2, ..., i + 1.

Adder / subtractor 1 of the value of X _k in register 21 via bus 105
7 to the input C of the register 18, and at the same time, to transfer the value of Y _k of the register 18 to the input D of the adder / subtractor 17 via the bus 104. At the same time, the barrel shifter 19 transfers the value of Y _k of the register 18 to The value is right-shifted by the number of double digits of k and transferred to the input B of the adder / subtractor 20. Next, the adder / subtractor 20 is controlled by the value popped from the stack 22, and the result is written in the register 21. At the same time, the adder / subtractor 17 is controlled by the value popped from the stack 22, and the result is shifted to the left by one digit and written in the register 18.

As a result, the mantissa part of cos θ is obtained in the register 21, and the mantissa part of sin θ is obtained in the register 18.

For the above-mentioned conventional examples, the following references are referred to.

* Industrial books: Hiroshi Hagiwara, "Microprogramming" * Reference patent: Takashi Nakayama, NEC Corporation "Trigonometric function calculation device" Japanese Patent Application No. 62-234195 [Problems to be solved by the invention] Conventional trigonometric functions described above In the arithmetic unit, in order to obtain the result of the trigonometric function calculation, a constant ROM of a constant Γ _k ,
There is a drawback that it is necessary to prepare an extra constant ROM of constant 1 / K, and recently, since it is necessary to perform high-precision arithmetic, the following problems have occurred. .

(1) In floating point arithmetic, the number of digits used is increasing in order to improve the precision of the mantissa part. Therefore, constant ROM
The number of digits of is increasing similarly.

(2) In order to make the domain of the function wide, the range of exponents that can be calculated becomes wide, and for this reason, it is necessary to increase the number of constant ROMs to be prepared.

In order to deal with the above problems, the conventional ROM
However, because of this, the amount of ROM hardware is increasing, and the proportion of constant ROM in the computer hardware is also increasing. Especially in a microprocessor, there is a drawback that a large amount of required area is consumed for a constant ROM, which causes an obstacle to implementation of other main hardware.

[Means for solving the problem]

The trigonometric function verification device of the present invention is configured with trigonometric functions sin θ and c
In a trigonometric function computing device intended to compute osθ, input an output value k of an exponent computing unit included in the computing device and compare the output value k with a predetermined constant. comparing means for matching the function ^{_{Γ k = 2 k × arctan (}} 2 -k) ( where, k = 0,1 ..., m- 1: m is a positive integer) values of gamma _k in, k from k = 0 = M−1, a constant generating unit of Γ _k that is sequentially generated as a constant value, (However, k = 0,1, ... m-1, m is a positive integer) The value of 1 / K is generated as a sequential constant value from k = 0 to k = m-1. The 1 / K constant generating means for selectively outputting the constant value of k according to the comparison result in the comparing means and the address of the constant generating means of Γ _k are
First adding means for selecting and incrementing according to the comparison result in the comparing means, and processing for adding 1 to the MSB of the output content of the constant generating means of Γ _k are performed according to the comparison result in the comparing means. And a second adding means for selectively executing.

EXAMPLES Next, the present invention will be described with reference to the drawings. First
The figure is a block diagram of one embodiment of the present invention. As shown in FIG. 1, in this embodiment, an exponent part computing part 1, a comparing part 2, constant ROMs 3 and 4, adders and 6, adder / subtractors 7 and 10, registers 8 and 11 are provided. , A barrel shifter 9, a stack 12, and a multiplier 13.

Before explaining the operation of the present embodiment, an outline of the operating principle of the present invention will be described. FIGS. 3 and 4 are tables showing the contents of the constant ROM of the constant Γ _k and the constant I / K, which are used when the algorithm is used in the above-mentioned conventional example. As is clear by comparing and comparing FIGS. 3 and 4, the constant ROM (constant γ _k
The difference between the contents when k = 19 in the figure) and the contents when k = 20 in the constant ROM of constant 1 / k is MSB (most signif
icant bit) is only different. Similarly, even when the value of k increases, the same thing can be said.

Therefore, it is understood that k = 0 to k = 18 should be prepared as the contents of the constant ROM of the constant 1 / K. when k becomes 19 or more, the MSB of the content of the constant ROM of 'constant gamma _k corresponding _to' k incrementing the value of the k "1"
The constant 1 / K can be obtained by adding.

Therefore, according to the present invention, the contents of the constant ROM of the constant 1 / K can be significantly reduced, the scale of the constant ROM can be reduced, the ROM hardware amount can be reduced, and the constant ROM of the computer hardware can be reduced. The ratio can be reduced.

As is clear from the comparison between FIG. 1 and FIG. 2, in this embodiment, the comparator 2, the adders 5 and 6 are used.
However, in practice, at the computer level capable of processing trigonometric functions, comparators, adders, etc. are provided for other purposes, and therefore their hardware may be diverted. Yes, there is no need to actually increase the hardware.

In FIG. 1, registers 11 and 8 are registers for storing two types of variables, an exponent arithmetic unit 1 is an arithmetic unit for performing exponential arithmetic, and a comparator 2 is an arithmetic result of the exponent arithmetic unit 1. By looking at it, it is judged whether k is a numerical value of 19 or more or less than 19, and a comparator or addition that decides whether to select the constant ROM4 of Γ _k or the constant ROM3 of constant 1 / K Bowl 6
Is the value of the constant Γ _k depending on the selection result in the comparator 2.
Adder that can add "1" to MSB, adder 5
Is an adder capable of incrementing the address of the constant ROM output from the exponent arithmetic unit 1, and the barrel shifter 9 can shift the contents of the register 8 right by the processing contents of the exponent arithmetic unit 1. Barrel shifter, constant ROM3
Is the constant 1 / K in the equation (13) (K = 0,1, ..., 18)
Is a 19-word constant ROM that stores the constants, and a constant ROM4 is an m-word constant ROM that stores the constant Γ _k in the equation (11).
2 is similar to the conventional example, the sequence {a _k } of First-In, Last-Ou
Stack that can be accumulated by t method, multiplier 13
Is a multiplier capable of inputting a multiplier and a multiplicand from the bus 101 and outputting the product of them to the bus 101. The bus 101 is a bus for transferring the input / output data and the intermediate result of the operation, the bus 102 is a bus for transferring the value of the register 11, and the bus 103 is a constant ROM required in the exponent calculator 1. It is a bus for transferring the address of.

The operation of this embodiment will be described below with reference to FIG.

A value of θ = 2− ⁱ × Θ (1 ≦ Θ <2, i is an integer) represented by a binary floating point number is input from the bus 101, and the exponent part i is input to the exponent part calculator 1. The mantissa part Θ of θ is input to the register 11.

The output value k of the exponent calculator 1 is incremented as k = i, i + 1, i + 2, ..., M-1 to repeatedly execute the following operation.

The V _k value stored in the register 11 is transferred to the input A of the adder / subtractor 10 via the bus 102, and at the same time, the address of the constant ROM obtained in the exponent calculator 1 is input to the adder / subtractor 5. , Addition is not performed in the adder / subtractor 5, and the address as it is is sent to the constant ROM 4 of the constant Γ _k . The value output from the constant ROM 4 is transferred to the adder 6, but no addition is performed here, and is transferred to the bus 101. At the same time, the value of the code digit of the register 11 is pushed onto the stack 12. Next, the operation of the adder / subtractor 10 is controlled depending on whether the sign digit of the register 11 is positive or negative.
Digit shift and write to register 11.

The V _m value is fetched from the register 11 via the bus 101 and transferred to the register 8 and the multiplier 13.

The exponent part calculator 1 transfers the value of k obtained from the value of the exponent part to the comparator 2 and determines whether the value of k is less than 19 or 19 or more.

If it is less than 19, the address of the constant ROM obtained from the exponent calculator 1 is directly converted to the constant 1 / K via the bus 103.
And sends the constant 1 / K to the bus 101.

If it is 19 or more, the address of the constant ROM obtained by the exponent calculator 1 is incremented by the adder 5 and sent to the constant ROM 4 of the constant Γ _k . Next, the adder 6 adds “1” to the MSB of the output value of the constant ROM 4 having the constant Γ _k , and transfers it to the bus 101.

In the above, the constant transferred to the bus 101 is transferred to the multiplier 13 via the bus 101, and the value of the product V _m / K is transferred to the bus 101.
To the register 11 via.

 The following operation is repeatedly executed while decrementing the output value k of the exponent calculator 1 to k = m, m-1, m-2, ..., I + 1.

The value of X _k of the register 11 is transferred to the input C of the adder / subtractor 7 via the bus 102, and at the same time, the value of Y _k of the register 8 is transferred to the input D of the adder / subtractor 7 via the bus 101. At the same time, in the barrel shifter 9, the Y _k value in the register 8 is right-shifted by the double digit number of k described above and transferred to the input B of the adder / subtractor 10. Next, the adder / subtractor 10 is controlled by the value popped from the stack 12, and the result is written in the register 11. At the same time, the adder / subtractor 7 is controlled by the value popped from the stack 12 and the result is shifted to the left by one digit and written in the register 8.

As a result, the mantissa part of cos θ is obtained in the register 11 and the mantissa part of sin θ is obtained in the register 8 at the same time.

〔The invention's effect〕

As described above in detail, according to the present invention, since the use area of the constant ROM of the constant 1 / K can be reduced, the hardware amount of the constant ROM is reduced as compared with the conventional trigonometric function operation device. And the constants in the computer hardware
The effect is that the proportion of ROM can be greatly reduced. Especially in microprocessors, the constant RO
The reduction of the area occupied by M has the effect of facilitating the mounting of other hardware.

[Brief description of drawings]

FIG. 1 is a block diagram of an embodiment of the present invention, and FIG. 2 is
The block diagram of the conventional example, FIG. 3 and FIG. 4 are views showing the contents of the constant ROM, respectively. In the figure, 1, 14 ... Exponent part calculator, 2 ... Comparator, 3,
4,15,16 …… Constant ROM, 5,6 …… Adder, 7,10,17,20 ……
Adder / subtractor, 8,11,18,21 …… register, 9,19 …… barrel shifter, 12,22 …… stack, 13,23 …… multiplier.

Claims (1)

(57) [Claims] 1. A trigonometric function computing device for computing trigonometric functions sin θ and cos θ, wherein an output value k of an exponential part computing unit included in the computing device is input to previously output the output value k. comparing means for comparing and collating a predetermined constant which is determined, the function ^{_{Γ k = 2 k × arctan (}} 2 -k) ( where, k = 0,1 ..., m- 1: m is a positive integer) gamma in _k Constant value generation means of Γ _{k that} generates the value of as a next constant value from k = 0 to k = m−1, (However, k = 0, 1, ... M-1, m is a positive integer) The value of 1 / K is generated as the next constant value from k = 0 to k = m-1. 1 / K constant generating means for selectively outputting the constant value of k according to the comparison result in the comparing means and the address of the constant generating means for Γ _k are selected according to the comparison result in the comparing means. First adding means for incrementing, and second adding means for selectively executing the processing of adding 1 to the MSB of the output content of the constant generating means of Γ _k according to the comparison result by the comparing means. A trigonometric function operation device comprising: