JPH04112320A - Trigonometrical function arithmetic device - Google Patents

Abstract

PURPOSE:To reduce the rate of the number of ROMs contained in the computer hardware by carrying out selectively the operations to add '1' to the output contents of a constant generating means based on the result of comparison of a comparator means. CONSTITUTION:A comparator 2 stores and compares the output value (k) of an exponent computing element 1 with a prescribed constant. The finction values are sequentially generated as the constants in a range covering K=0 through k=m-1 in rK=2<k> X arctan (2<-k>) where k=0, 1...m-1 and (m) is equal to an integer. At the same time, the values of 1/K are sequentially generated as constants in a range covering k=0 through k=m-1 in the function rK=II<m> (1+1<-2k>)<1/2> where k=0, 1...m-1 and (m) is a positive integer. The constant values of 1/k are selectively outputted based on the result of comparison of the comparator 2. Then an adder 6 selects the addresses of rK in response to the result of comparison of the comparator 2 and increases these addresses. An adder 10 adds selectively '1' to the most significant bit MSB of the output contents of the function rK based on the result of comparison of the comparator 2. As a result, the occupying rates of the constant ROM 3 and 4 are reduced in the computer hardware.

Description

[Detailed description of the invention] [Industrial application field] The present invention relates to a trigonometric function calculation device.

[Conventional technology]

Conventionally, the method of calculating trigonometric functions is to expand the Tiller series of the following equation, ie, sin θ = θ ~ θ3/3! Mo θ5151-θ7/7! +
. . . (1) or a method of obtaining by rational function approximation using continued fraction expansion, Tievishev series expansion, etc. is used.

However, both of the above methods require a large number of multiplications or divisions, resulting in a long calculation time and an unsatisfactory calculation speed.

As a solution to the above problems, C
ORD I C (Cordina Le Rojat, i
onDigital Compute, er). The C0RD IC algorithm is known to be effective even in computers that are not equipped with high-speed multipliers, since operations can be performed by addition, subtraction, and right shift.

The calculation principle of the C'0RDIC algorithm is as follows:
The summary is as follows.

When calculating sin θ and cos θ with precision of n (positive integer) digits in binary system, the angle θ is the constant γ1 and the sequence 1
It is expressed by the following equation using a and l.

θ=aoXγo+alXγ1−a2Xγ2+...
・+ a, -I Xγn-1+ε・・・・・・・・・(
2) However, γ, = arctan(2''-')...
・・・・・・・・・−・・・−・・・(3)ak”i÷1.
1i ・・・・・・・・・・・・・・・(4) Sequence 1a5) is a method similar to division in the separation method,
- Can make decisions at will. Furthermore, this number sequence 1ak
The process of determining l is called pseudo-division.

Therefore, sinθ and cosθ are Rm=1
/' (L + 2-")"2 ・・・-・・・・・・
(5), it can be obtained as follows using the addition theorem.

When ak=+1, +1 to φ =φ5+γ3cos(
+ to φ)=Rk(cosφy −2−′ X si
nφk)・・・・・・−・・・・・・・・・(6) sin
(++ to φ) = Rk (+2-' to sinφ) cosφ
k)・・・－・・・・・・・・・・・・ ku7)ak=
1, φI<+1 = φh 7kcos (
+ to φ) = Rm (+2-' to cosφ
nφk)・・・・・・・・・・・・・・・・・・(8) sin
(++ to φ)=Rm(sinφh 2−”X co
sφk)...--(9>) Here, by using Ro as the initial value and repeatedly performing the above calculation operations to increase the value of k, φ5 can be obtained. θ is approached. Through such operations, sin θ and cos θ are obtained. This calculation process is called pseudo multiplication.

The principle of floating point arithmetic of triangular numbers using the C0RDIC algorithm will be explained below using formulas.

[Let θ (0<θ<π/2) be the angle that corresponds to 11 people.

[2] The above angle θ is set as θ=2−'Xe (1≦θ×2, i is an integer), and i
(exponent part) and θ (mantissa part) are determined (floating point conversion)
Then, let V+=e be the initial value.

[3] Repeat the following [41] for k=1, i+1, i + 2, -1m-1 (m is a positive integer).

[41 If Vk≧0, ak=+1 ■5 If 0, then ak=1, Vk+4=2X
(VkamXr"k) - (10) However, r' k= 2'-' X arctan (2-')
---(11) [5j x-= V=X (1/K)
,Y,=V,−,,111111,,,,,,,,,
, , (12) is the initial value. Here, is a constant that satisfies .

E61 k=m, m-1, m-2,..., i + 1
, repeat [7] below.

[7] Xh-x=L-a, x2-2kx:yk"""
'-"""'<14) Yk-1= (Ym f akX
L)/2−−− ・= ・−・−(15) fLI
sin θ:H21X Y , cos θ=X, are obtained at the same time.

When the above C0RD IC algorithm is used in a computer, the constant of the above equation (11) 1 equation % (2) and the above equation (13) K = JT ”(1+ 2-2k>”2(k=0 ,1
．． ..., n) A constant for 1/, which is the reciprocal of , is required.

On the other hand, when using this algorithm, the range of θ is
It is applied only when O x θ x π/4, so when π,·4 or more, the remainder is taken from the input categor by π/4, and the present algorithm is used for that value. Therefore,
The data input to this algorithm has an exponent part ranging from -m to -1.

This means that π,/4=3.14・−・−/4=0.785・−=1
．． This is clear from the fact that it is 57·-X2-1.

As mentioned above, when the exponent part changes from -m to -1, m constant values are also required. That is, the above (11)
The constant ``,'' in the equation, and the constant 1/ obtained from equation (13) must be prepared in m pieces each.In conventional calculators, the constants are stored in advance in a constant ROM (Read
0nly 14emory) and is used for calculation processing of trigonometric functions. An example of the contents of the constant ROM for "k" in this case is shown in the table of FIG.
An example of the contents of the constant ROM in / is shown in FIG. However, this is a table showing constants when the calculation range of the mantissa part is from 32 to -1 (decimal number), and is shown in hexadecimal notation.

A conventional trigonometric function calculation device using the above C0RDIC algorithm is configured as shown in FIG. 2, for example.

As shown in FIG. 2, in this conventional example, the exponent part calculator 1
4, constant ROM 15 and 16, adder/subtractor I7 and 20, registers 18 and 21, barrel register 19,
It is configured to include a stack 22 and a multiplier 23. The operation will be explained below with reference to FIG.

■ Input the value of θ=2−'
Enter. The mantissa part e of θ is input to the register 21.

(2) While incrementing the output value k of the exponent part calculator 14 as follows: k=i, i+l, i+2, -1m-1, the operation shown in (2) below is repeatedly executed.

■ Transfer the Yk value stored in register 21 to bus 10.
5 to the input A of the adder/subtractor 20, and at the same time, the value in the constant ROM 16 of the constant Γk selected by the address obtained in the exponent part calculator 14 is transferred to the input B of the adder/subtractor 20 via the bus 104. Forward. At the same time, depending on the sign digit of the register 21, the adder/subtractor 2
1 and shifts its output by one digit to the left and writes it into the register 21.

(2) Take out the value v from the register 21 via the bus 104 and transfer it to the register 18 and multiplier 23. The value in the constant ROM 15 for the constant 1/ selected by the address obtained by the exponent calculator 14 is transferred to the bus 104.
The product V, /K of Vyo and L/ni obtained in the multiplier 23 is transferred to the register 21 via the bus 104.

■ The output value k of the exponent part calculator 14.

While decrementing k=m, m-1, m-2, -..., i + 1, the following operation (2) is repeatedly executed.

■ Transfer the value of Xk of the register 21 to the input C of the adder/subtractor 17 via the bus 105, and at the same time transfer the value of Yk of the register 18 to the input of the adder/subtractor 17 via the bus 104; At the same time, in the barrel shifter 19, Yk in the register 18 is shifted to the right by twice the number of digits of the aforementioned k and transferred to the input B of the adder/subtracter 20. and writes the result to the register 21. At the same time, the value popped from the stack 22 causes the adder/subtractor L7 to vlm, and the result is shifted to the left by one digit and written into the register 21.

(2) As a result of the above, the register 21 obtains the mantissa part of cos θ, and the register 18 obtains the mantissa part of sin θ.

In addition, regarding the above-mentioned conventional example, the following references etc. are referred to.

*Industrial book: ``Microprogramming'' written by Hiroshi Hagiwara, patent application 1988-
No. 234195 [Problem to be Solved by the Invention] In the above-mentioned conventional trigonometric function calculation device, in order to obtain the result of trigonometric function calculation, a constant ROM of constant ``2'',
There is a drawback that it is necessary to prepare an extra constant ROM for the constant 1/, and recently there is a need to perform high-precision calculations, so the following problems have arisen. .

(1) In floating point arithmetic, the number of digits used is increasing in order to improve the precision of the mantissa. Therefore, the constant RO
The number of digits of M is also increasing in the same way.

(2) In order to widen the domain of the function, the range of exponents that can be operated on becomes wider, and therefore it is necessary to increase the number of constant ROMs to be prepared.

In order to deal with the above problems, conventionally, ROM
However, as a result, the amount of ROM hardware increases, and the proportion of constant ROM in the computer hardware also tends to increase. Particularly in microprocessors, a disadvantage is that the large amount of area required for constant ROM is a hindrance to the implementation of the main hardware of the microprocessor.

[Means to solve the problem]

The trigonometric function calculation device of the present invention is a trigonometric function calculation device whose purpose is to calculate trigonometric functions sin θ and cos θ. a comparison means for comparing and comparing the value k with a predetermined constant; and a function r', =2'X arcjan(2-k) (where k=0, 1-, m-1: m is positive). Γk at (integer)
, which sequentially generates the value of as a constant value from k=o until =m-1, and a function =IT'''(1+2-2k)l/2 (where k
= O1L', -・-m -1, m is a positive integer) is successively generated as a constant value until = m-1 by k=o, and the constant value of this 1/ is a constant generating means for selecting and outputting Γk according to the comparison result in the comparing means; and a second addition means that selectively executes the process of adding 1 to the MSB of the output content of the constant generation means for k according to the comparison result in the comparison means. .

〔Example〕

Next, the present invention will be explained with reference to the drawings. FIG. 1 is a block diagram of one embodiment of the present invention. As shown in FIG. 1, this embodiment includes an exponent calculation unit l, a comparator 2, constant ROMs 3 and 4, adders 5 and 6,
It is configured to include adder/subtractors 7 and 10, registers 8 and 11, barrel shifter 9, stack 12, and multiplier 13.

Before explaining the operation of this embodiment, the outline of the operating principle of the present invention will be explained. 3 and 4 are tables showing the contents of the constant ROM for the constant "k" and the constant 17, respectively, which are used when using the algorithm in the conventional example described above. As is clear by comparing and contrasting the constant ROM of the constant ``,
The difference between the content when N = 19 in (Fig. 3) and the content when N = 20 in the constant ROM of constant 1/ is as follows.
MSB (most 51gn1ficant bit
) are different. Similarly, the same thing can be said even when the value of k increases.

Therefore, the contents of the constant ROM for the constant 1/ are l<
It turns out that it is sufficient to prepare from =Q to =18. When k becomes 19 or more, by incrementing the value of k and adding "1" to the MSB of the constant ROM contents of the constant Fk corresponding to ', the constant 1/
It becomes possible to obtain K.

Therefore, according to the present invention, it is possible to significantly reduce the content of the constant ROM in constant 1/, the scale of the constant ROM is reduced, the amount of ROM hardware is also reduced, and the constant ROM occupies less space in the computer hardware. The ratio can be reduced.

Furthermore, as is clear from the comparison between Figures 1 and 2,
In this embodiment, a comparator 2 and adders 5 and 6 are added, but in reality, at the computer level capable of trigonometric function processing, the comparator, adder, etc.
Since the hardware is provided for other purposes, it is possible to reuse the hardware, and there is no need to actually increase the hardware.

In FIG. 1, registers 11 and 8 are registers that store variables of class 2, exponent part arithmetic unit 1 is an arithmetic unit that performs exponent part arithmetic, and comparator 2 is the calculation result of exponent part arithmetic unit 1. A comparator that determines whether k is a value between 19 and E or a value that is not equal to 19, and selects either constant ROM4 for Γk or constant ROM3 for constant 17; The adder 6 uses the selection result in the comparator 2 to
Adder 5, which is an adder that can add "1" to the MSB of the value of constant "k," is an adder that can increment the constant ROM address output from exponent part calculator 1. The barrel shifter 9 is a barrel shifter that can shift the contents of the register 8 to the right according to the processing contents of the exponent part calculator 1, and the constant ROM 3 is a constant 1/K (k・0.l, . . . in equation (13)). . . . 18) The constant ROM 4 is a 19-word constant ROM that stores the constant "3" in the equation (11), and the stack 12 is a constant ROM of m words that stores the constant "3" in the formula (11). l'(Fir
The multiplier 13 is a multiplier that can input a multiplier and a multiplicand from the bus 101 and output their product to the bus 101. The bus 101 is a bus for transferring input/output data and intermediate results of operations;
The bus 102 is a bus for transferring the value of the register 11, and the bus 103 is a bus for transferring the address of the constant ROM determined by the exponent part calculator 1.

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

(2) A value of θ=2-' The mantissa part θ of θ is input to the register 11.

(2) While incrementing the output value k of the exponent part calculator 1 as follows: k=i, i+l, i+2, . . . , m, -1, the operation shown in (2) below is repeatedly executed.

■ The vk value stored in register 11 is transferred to bus 10.
2 to the human power A of the adder IO, and at the same time,
The address of the constant ROM obtained in the exponent part calculator 1 is input to the adder/subtracter 5, but no addition is performed in the adder/subtracter 5, and the address as it is is input to the constant R of the constant Γk.
Send to OM4. The value output from constant ROM4 is
The data is transferred to the adder 6, but no addition is performed here either, and the data is transferred to the bus 101. At the same time, the value of the sign digit of the register 11 is pushed onto the stack 12.The operation of the adder/subtractor IO is controlled depending on the sign digit of the register 11, and its output is shifted to the left by 1 to the register 12.
Write to 1.

(2) Take out the value V from the register 11 via the bus 101 and transfer it to the register 8 and the multiplier 13;

■ The value of k determined by the value of the exponent part is transferred from the exponent part calculator 1 to the comparator 2, and the value of k is 19.
It is determined whether it is less than 19 or 19 or more.

If it is less than 19, the address of the constant ROM obtained from the exponent part calculator 1 is directly sent to the constant ROM 3 of the constant 1/ via the bus 103, and the constant L/K is sent to the constant ROM 3 of the constant 1/K via the bus 103.
Transfer to 1.

If it is 19 or more, the address of the constant ROM obtained from the exponent part calculator 1 is incremented in the adder 5 and sent to the constant ROM 4 of the constant Γk.
The adder 6 calculates the MS of the output value of the constant ROM 4 of constant ``2''.
“1” is added to B and transferred to the bus 101.

■ In above ■, the constant transferred to bus 101 is
The value of the product V, , / is transferred to the register 11 via the bus 101.

(2) Repeat the operation (2) below while decrementing the output value k of the exponent calculator 1 as follows: = m, m -1, m-2, . . . , i+1.

■ Transfer the value of xk in register 11 to input C of adder/subtractor 7 via bus 102, and at the same time transfer the value of xk in register 8 to input C of adder/subtractor 7.
is transferred to the input of the adder/subtractor 7 via the bus 101, and at the same time, in the barrel shifter 9, the value is transferred to the register 8.
The Yk value of is shifted to the right by twice the number of digits of the aforementioned k, and is 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 to the register 11.
The adder/subtractor 7 is controlled by the value popped from 2, and the result is shifted to the left by one digit and written to the register 8.

(2) As a result of the above, 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.

〔Effect of the invention〕

As described above in detail, the present invention can reduce the area used in the constant ROM by the constant 1/;
Compared to conventional trigonometric function calculation devices, the hardware amount of the constant ROM can be reduced, and the proportion of the constant ROM in the computer hardware can be significantly reduced. Especially in microprocessors,
By reducing the area occupied by the constant ROM, there is an effect that other hardware can be easily implemented.

[Brief explanation of drawings]

FIG. 1 is a block diagram of an embodiment of the present invention, and FIG. 2 is a block diagram of an embodiment of the present invention.
The block diagrams of the conventional example, FIG. 3 and FIG. 4, each show the contents of the constant ROM. In the figure, 1, 14... exponent part calculator, 2...
......-Comparator, 3.4.1.5.16...
...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)

[Scope of Claims] A trigonometric function calculation device whose purpose is to calculate trigonometric functions sin θ and cos θ, wherein an output value k of an exponent part calculation unit included in the calculation device is input, and the output value k is calculated. Comparison means for comparison with a predetermined constant, and function Γ_k=2^k×ar
ctan (2^-^k) (where k=0, 1..., m
-1: m is a positive integer) is used to generate the value of Γ_k as a constant value sequentially from k=0 to k=m-1, and ▲Mathematical formulas, chemical formulas, tables, etc.▼ (However, , k=0, 1,...m-1, where m is a positive integer) is successively generated as a constant value from k=0 to k=m-1, and this The constant value of 1/k is
1 for selectively outputting according to the comparison result in the comparison means;
/K constant generation means; first addition means for selecting and incrementing the address of the Γ_k constant generation means according to the comparison result in the comparison means; and MSB of the output content of the Γ_k constant generation means. A trigonometric function calculation device comprising: second addition means for selectively executing a process of adding 1 to , depending on a comparison result in the comparison means.