CN103150137A - Implementation method of single-precision floating point trigonometric function covering full circumferential angle - Google Patents

Abstract

The invention discloses an implementation method of single-precision floating point trigonometric function covering full circumferential angle, and belongs to the field of processing of digital signal. The method comprises following steps: firstly, a preprocessing module CORDIC-PRE receives input single-precision floating point data, records the quadrant information of original data, converts the single-precision floating point data within a set angle range into high-precision floating point data, and inputs the high-precision floating point data to an iterative operation module CORDIC-CORE; secondly, the CORDIC-CORE finishes iterative operation of the CORDIC algorithm to the input data by adopting high-precision floating point operation; the result is input to a postprocessing module CORDIC-POST; thirdly, the CORDIC-POST performs quadrant recovery to sine and cosine functional values to be required to be calculated as per the quadrant information recorded in the CORDIC-PRE aiming to the input data; and the data after being recovered is converted into the precision floating point data and output. The implementation method is suitable for actual operation of CORDIC algorithm.

Description

A kind of implementation method that covers the single-precision floating point trigonometric function of circle angle

Technical field

The invention belongs to digital processing field, be specifically related to a kind of implementation method of small size single-precision floating point trigonometric function.

Background technology

Precision and dynamic range that the modern digital signal is processed computing have proposed more and more higher requirement, and the single-precision floating point trigonometric function operation is also more and more used.Rotation of coordinate digital computation (CORDIC) is started with from computing itself, and the trigonometric function operation of complexity is decomposed into simple plus-minus method and shift operation, makes trigonometric function operation be easy to realize on hardware.

At present, a lot of research is also done in the realization of single-precision floating point trigonometric function both at home and abroad, the needs according to oneself on the basis of classical cordic algorithm have carried out some improvement, and it is applied in actual operation.

Yet for the single-precision floating point trigonometric function, it is two shortcomings of main existence in application, the one, and the angular coverage of classical cordic algorithm is inadequate, can only reach [99.88 °, 99.88 °]; Two cordic algorithms that are based on Float Point Unit can take a lot of logical resources, and area occupied is large, frequency of operation is not high.

Therefore how to make cordic algorithm can be applicable to the angle value of input arbitrarily, and the larger shortcoming of area occupied that occurs in improvement computation process is problem demanding prompt solution.

Summary of the invention

In view of this, the invention provides a kind of implementation method of small size single-precision floating point trigonometric function, the angular coverage that has solved classical cordic algorithm not and logical resource take too much problem, enlarged angular coverage, effectively reduce logical resource, improved frequency of operation.

To achieve the above object of the invention, technical scheme of the present invention is: comprise the steps:

Step 1, pretreatment module CORDIC_PRE receive the single-precision floating-point data of input, adopt following method single-precision floating-point data to be transformed into [π 4, π 4] in scope, and be converted to the high precision fixed-point data, the high precision fixed-point data that obtains is inputed to interative computation module CORDIC_CORE;

If the input data are the angle value θ of the single-precision floating-point data form of any range, minute following two kinds of situations are carried out the angular range conversion, and the θ after changing is by being converted to single-precision floating-point data the high precision fixed-point data;

If θ is in the scope of [2 π, 2 π], θ is rotated take pi/2 or π as unit, rotation to [π 4, and π 4] scope, the angle that record rotates;

If θ is in (2 π, ∞) ∪ (∞,-2 π) in scope, θ is added or deducts 2n π, n is integer, make θ be converted to [2 π, 2 π] within scope, the θ after conversion is rotated take pi/2 or π as unit again, rotation is to [π 4, π 4] in scope, the angle that record rotates;

If the input data are the vector (x, y) of any range, (x, y) transformed to radius is in 1 circumference, obtain vector (x ＇, y ＇), then with vector (x ＇, y ＇) rotation to [π 4, and π 4] circumference range, is recorded the angle of rotating; Single-precision floating-point data is converted to the high precision fixed-point data;

Step 2, CORDIC_CORE complete the cordic algorithm interative computation to the input high-precision fixed point processing of the data wherein; Result is inputed to post-processing module CORDIC_POST;

Step 3, CORDIC_POST are processed according to different situations for to input data wherein:

If need to calculate the sin cos functions value, input to and be the sine value of same angle and cosine value in CORDIC_POST, in CORDIC_POST according to the angle of rotating that records in CORDIC_PRE, carry out data according to the transformation relation of sin cos functions value and recover;

Wherein the method for data recovery is as follows:

If the anglec of rotation that records is-pi/2, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}-\mathrm{\π}/2)=-\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}-\mathrm{\π}/2)=\sin \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the cosine function value of former angle value, and the cosine function value that inputs in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is pi/2, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}+\mathrm{\π}/2)=\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}+\mathrm{\π}/2)=-\sin \mathrm{\θ}\end{array},$ The sine function that namely inputs in CORDIC_POST is the cosine function value of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is ± π, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}\±\mathrm{\π})=-\sin \mathrm{\θ}\\ \cos (\mathrm{\θ}\±\mathrm{\π})=-\cos \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the sine function of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the cosine function value of former angle;

If need to calculate angle value, according to the angle of rotating that records in CORDIC_PRE, the angle value that is input in CORDIC_POST recovered;

Data-switching after recovering is become accuracy floating-point data and output.

Further, in step 1, the concrete grammar of the angle that record rotates is:

Divide as follows angular interval: [π 4, and π 4] as the first interval, (π 4,3 π 4] be between Second Region, (3 π 4 ,-3 π 4] be the 3rd interval, (3 π 4 ,-π 4) are the 4th interval

If the input data are the angle value θ of the single-precision floating-point data form of any range, to not belong to [2 π, 2 π] scope in θ be converted to [2 π, 2 π] in scope, be in [2 π for θ, 2 π] scope in situation, only record the angle that the residing angular interval of θ can obtain rotating; If same input data are the vector (x, y) of any range, will (x, y) transform to radius and be in 1 circumference, obtain vector (x ＇, y ＇), only record the angle that (x ＇, y ＇) residing angular interval can obtain rotating;

If angle θ or vector (x ＇, y ＇) are in the first interval, the angle of rotating is zero;

If angle θ or vector (x ＇, y ＇) are between Second Region, the angle of rotating is-pi/2;

If angle θ or vector (x ＇, y ＇) are in the 3rd interval, the angle of rotating is ± π;

If angle θ or vector (x ＇, y ＇) are in the 4th interval, the angle of rotating is+pi/2.

Beneficial effect

(1), the present invention's mode of adopting mathematics to change, any range input is transformed in the angular range that cordic algorithm can cover, therefore indirectly enlarge angular coverage, can carry out computing to the input data of any range; The present invention adopts the unit realization of high-precision fixed point processing simultaneously, has greatly reduced logical resource, has improved frequency of operation.

(2), the present invention is owing to adopting pipeline organization to realize interative computation in cordic algorithm, than the existing single-precision floating point trigonometric function operation unit institute feedback arrangement of employing usually, its data throughout is large, is more suitable for high efficiency operation.

Description of drawings

Fig. 1 is the hardware structure diagram of single-precision floating point trigonometric function operation; In figure, clk is the input end of clock of module, rst is the reset instruction input end of module, x_in, y_in, z_in are the single-precision floating-point data input of cordic algorithm, x(0), y(0), z(0) be through the conversion after the high precision fixed-point data, first step input as the cordic algorithm iterative computation, x(n-1), y(n-1), z(n-1) be the final output of cordic algorithm iterative computation, x_out, y_out, z_out are the result output of final single-precision floating-point data form of the present invention;

Fig. 2 is angle of circumference interval division figure;

Fig. 3 is the hardware configuration schematic diagram of interative computation unit; In figure, x _i, y _iBe the i of the iteration input in step, x _i+1, y _i+1Be the i of the iteration output in step, σ _iIt is the i iteration coefficient in step;

Fig. 4 pipeline organization schematic diagram.

Embodiment

Elaborate below in conjunction with the embodiment of accompanying drawing to the inventive method.

Embodiment 1

The implementation method of a kind of small size single-precision floating point of the present invention trigonometric function, the method adopts the system that is comprised of pretreatment module (CORDIC_PRE), interative computation module (CORDIC_CORE) and post-processing module (CORDIC_POST) to realize, concrete system architecture as shown in Figure 1.Concrete steps are as follows:

Step 1, pretreatment module CORDIC_PRE receive the single-precision floating-point data of input, adopt following method single-precision floating-point data to be transformed into [π 4, π 4] circumference range in, and be converted to the high precision fixed-point data, the high precision fixed-point data that obtains is inputed to interative computation module CORDIC_CORE.

Because the angular coverage of classical cordic algorithm is inadequate, can only reach [99.88 °, 99.88 °], consider the precision of calculating, the present invention selects single-precision floating-point data to be transformed in the circumference range of [π 4, and π 4].

And for the single-precision floating-point data of inputting, the unsteady complexity increase that makes computing due to its radix point, and in fixed-point data, radix point position in number immobilizes, therefore need not consider again the decimal problem in computing, therefore transfer single-precision floating-point data to the high precision fixed-point data, also improved the efficient of algorithm when significantly reducing logical resource.

For traditional cordic algorithm, its sin cos functions adopts angle rotation to realize, the input data are the angle value θ of any range, and θ is single-precision floating-point data, and the form of single-precision floating-point data is (1) ^s* 1.f * 2 ^E-127, s is 1 bit sign position, and e represents 8 indexes, and f represents 23 significance bits; In cordic algorithm, arctan function is to adopt vector to realize, its input be the vectorial coordinate value (x, y) of any range, so this step need to consider that when changing the input data are θ and (x, y) two kinds of situations.

If the input data are the angle value θ of any range:

For the angle value θ outside [2 π, 2 π] scope, carrying out sin cos functions when calculating, it can be added or deducts 2n π, n is integer, make within it is converted into [2 π, 2 π] scope, and on the calculated value of the sine and cosine of θ without impact; And for the angle value θ within [2 π, 2 π] scope, if it is rotated take pi/2 or π as unit, the conversion of its sin cos functions value also has following rule to follow:

If therefore single-precision floating-point data will be transformed in the circumference range of [π 4, and π 4], consider to take following methods that the angle value θ in any range is changed, the θ after conversion is by being converted to single-precision floating-point data the high precision fixed-point data:

If θ is in [2 π, 2 π] scope, θ is rotated take pi/2 or π as unit, rotation to [π 4, and π 4] scope, the angle that record rotates.

If θ is in (2 π, ∞) ∪ (∞,-2 π) in scope, θ is added or deducts 2n π, n is integer, makes θ be converted to [2 π, 2 π] within scope, θ after conversion again take pi/2 or π as unit is rotated, is rotated to the set angle scope angle that record rotates.

If the input data are the vector (x, y) of any range:

Adopting following formula that (x, y) transformed to radius is in 1 circumference, obtains (x ＇, y ＇):

$\left\{\begin{array}{c}{x}^{\′}=x/\sqrt{x^2+y^2}\\ y^{\′}=y/\sqrt{x^2+y^2}\end{array}\right.$

Vector (x ＇, y ＇) for belonging in circumference records its block information in circumference, take pi/2 or π as unit is rotated, rotates in [π 4, and π 4] circumference range the angle that record rotates.Single-precision floating-point data is converted to the high precision fixed-point data.

Step 2, CORDIC_CORE complete the cordic algorithm interative computation to the input high-precision fixed point processing of the data wherein; Result is inputed to post-processing module CORDIC_POST.

Data input to interative computation module CORDIC_CORE, and the operating structure of CORDIC_CORE as shown in Figure 3.CORDIC_CORE adopts high-precision fixed point processing to complete the cordic algorithm interative computation.

Single-precision floating point computing meeting takies a large amount of logical resources, and the present invention is owing in step 1, the single-precision floating-point data of input being changed, therefore the present invention adopts high-precision fixed point processing unit can realize addition and shift operation in interative computation, completes the interative computation of cordic algorithm.Because of the shared logical resource in high-precision fixed point processing unit less, therefore interative computation can adopt pipeline organization to realize, pipeline organization as shown in Figure 4, the feedback arrangement that uses than the single-precision floating point arithmetic element, therefore pipeline organization can improve operation efficiency because data throughout increases.

If the data by pretreatment module CORDIC_PRE input are an angle value, the result that is finally drawn by CORDIC_CORE is sine value and the cosine value of this angle, inputs to post-processing module CORDIC_POST; If the data by pretreatment module CORDIC_PRE input are a vector value, the result that is finally drawn by CORDIC_CORE is the corresponding angle value of this vector, inputs to post-processing module CORDIC_POST.

Step 3, CORDIC_POST are processed according to different situations for to input data wherein:

If need to calculate the sin cos functions value, in CORDIC_POST according to the angle of rotating that records in CORDIC_PRE, carry out data according to the transformation relation of sin cos functions value and recover;

The concrete grammar that data are recovered is as follows:

If the anglec of rotation that records is-pi/2, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}-\mathrm{\π}/2)=-\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}-\mathrm{\π}/2)=\sin \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the cosine function value of former angle value, and the cosine function value that inputs in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is pi/2, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}+\mathrm{\π}/2)=\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}+\mathrm{\π}/2)=-\sin \mathrm{\θ}\end{array},$ The sine function that namely inputs in CORDIC_POST is the cosine function value of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is ± π, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}\±\mathrm{\π})=-\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}\±\mathrm{\π})=-\sin \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the sine function of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the cosine function value of former angle.

If need to calculate angle value, according to the angle of rotating that records in CORDIC_PRE, the angle value that obtains recovered.

The recovery of angle value is fairly simple, only carries out reverse rotation according to the angle of rotating of record and can recover.

Embodiment 2

The present embodiment still adopts the system that is comprised of pretreatment module (CORDIC_PRE), interative computation module (CORDIC_CORE) and post-processing module (CORDIC_POST) to realize.

Step 1, if pretreatment module CORDIC_PRE receives the single-precision floating-point data of input, adopt following method single-precision floating-point data to be transformed into [π 4, π 4] circumference range in, and be converted to the high precision fixed-point data, the high precision fixed-point data that obtains is inputed to interative computation module CORDIC_CORE.

If the input data of pretreatment module CORDIC_PRE are the angle value θ of the single-precision floating-point data form of any range, to not belong to [2 π, 2 π] scope in θ add or deduct 2n π, n is integer, makes within θ is converted to [2 π, 2 π] scope, be in [2 π for θ, 2 π] scope in situation, record the residing angular interval of θ, specific as follows:

According to mode demarcation interval as shown in Figure 4, namely [π 4, and π 4] as the first interval, (π 4,3 π 4] be between Second Region, (3 π 4 ,-3 π 4] be the 3rd interval, (3 π 4 ,-π 4) are the 4th interval.

So divide angular interval between with Second Region～when the rotation of four-range angle value is interval to first, by the residing angular interval of θ, according to following rule with the θ rotation to [π 4, and π 4] scope:

In if angle θ is between Second Region, the angle of the direction rotation pi/2 that it is reduced to angle value can enter in [π 4, and π 4] scope, and postrotational angle is θ-pi/2;

If angle θ is in the 3rd interval, the angle with its direction rotation π that no matter increases to the direction that reduces to angle value or to angle value all can enter in [π 4, and π 4] scope, and postrotational angle is θ ± π;

If angle θ is in the 4th interval, its angle to the direction rotation pi/2 of angle value increase can be entered in [π 4, and π 4] scope, postrotational angle is θ+pi/2;

And if angle value θ namely was in the first interval originally, need not rotation.

In like manner as can be known, if pretreatment module CORDIC_PRE input data are the vector (x, y) of any range, with (x, y) transforming to radius is in 1 circumference, obtain vector (x ＇, y ＇), record (x ＇, y ＇) residing circumference angular interval, and according to above-mentioned rule, (x ＇, y ＇) rotated to the circumference range of [π 4, and π 4].

Above single-precision floating-point data is converted to the high precision fixed-point data, and the high precision fixed-point data that obtains inputs to interative computation module CORDIC_CORE

Step 2, CORDIC_CORE complete the cordic algorithm interative computation to the input high-precision fixed point processing of the data wherein; Result is inputed to post-processing module CORDIC_POST.

Step 3, CORDIC_POST are processed according to different situations for to input data wherein:

If need to calculate the sin cos functions value, in CORDIC_POST according to the block information that records in CORDIC_PRE, carry out data according to the transformation relation of sin cos functions value and recover;

The concrete grammar that data are recovered is as follows:

If the block information that records is between Second Region, corresponding angle of rotating is-pi/2; According to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}-\mathrm{\π}/2)=-\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}-\mathrm{\π}/2)=\sin \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the cosine function value of former angle value, and the cosine function value that inputs in CORDIC_POST is the sine function of former angle;

If the block information that records is the 4th interval, corresponding angle of rotating is+pi/2, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}+\mathrm{\π}/2)=\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}+\mathrm{\π}/2)=-\sin \mathrm{\θ}\end{array},$ The sine function that namely inputs in CORDIC_POST is the cosine function value of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the sine function of former angle;

If the block information that records is the 3rd interval, corresponding angle of rotating is ± π, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}\±\mathrm{\π})=-\sin \mathrm{\θ}\\ \cos (\mathrm{\θ}\±\mathrm{\π})=-\cos \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the sine function of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the cosine function value of former angle.

If need to calculate angle value, infer institute's anglec of rotation according to the block information that records in CORDIC_PRE, with the angle of being rotated, the angle value reverse rotation that obtains is recovered.

Data-switching after recovering is become single-precision floating-point data and output.

In sum, these are only preferred embodiment of the present invention, is not for limiting protection scope of the present invention.Within the spirit and principles in the present invention all, any modification of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (2)

1. an implementation method that covers the single-precision floating point trigonometric function of circle angle, is characterized in that, comprises the steps:

Step 1, pretreatment module CORDIC_PRE receive the single-precision floating-point data of input, adopt following method single-precision floating-point data to be transformed into [π/4, π/4] in scope, and be converted to the high precision fixed-point data, the high precision fixed-point data that obtains is inputed to interative computation module CORDIC_CORE;

If the input data are the angle value θ of the single-precision floating-point data form of any range, divide following two kinds of situations to carry out the angular range conversion, and the θ after changing realize that by the exponential part of single-precision floating-point data being appointed as 127 single-precision floating-point data arrives the conversion of high precision fixed-point data;

If θ is in the scope of [2 π, 2 π], θ is rotated take pi/2 or π as unit, rotation to [π/4, π/4] scope, the angle that record rotates;

If θ is in (2 π, ∞) ∪ (∞,-2 π) in scope, θ is added or deducts 2n π, n is integer, make θ be converted to [2 π, 2 π] within scope, the θ after conversion is rotated take pi/2 or π as unit again, rotation is to [π/4, π/4] in scope, the angle that record rotates;

If the input data are the vector (x, y) of any range, (x, y) transformed to radius is in 1 circumference, obtain vector (x ＇, y ＇), then with vector (x ＇, y ＇) rotation to [π/4, π/4] circumference range, is recorded the angle of rotating; Single-precision floating-point data is converted to the high precision fixed-point data;

Step 2, CORDIC_CORE complete the cordic algorithm interative computation to the input high-precision fixed point processing of the data wherein; Result is inputed to post-processing module CORDIC_POST;

Step 3, CORDIC_POST are processed according to different situations for to input data wherein:

If need to calculate the sin cos functions value, input to and be the sine value of same angle and cosine value in CORDIC_POST, in CORDIC_POST according to the angle of rotating that records in CORDIC_PRE, carry out data according to the transformation relation of sin cos functions value and recover;

The method that described data are recovered is as follows:

If the anglec of rotation that records is-pi/2, according to sin cos functions transformation relation s $\begin{array}{c}\sin (\mathrm{\θ}-\mathrm{\π}/2)=-\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}-\mathrm{\π}/2)=\sin \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the cosine function value of former angle value, and the cosine function value that inputs in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is pi/2, according to sin cos functions transformation relation s $\begin{array}{c}\sin (\mathrm{\θ}+\mathrm{\π}/2)=\cos \mathrm{\θ}\\ \cos (\mathrm{\θ}+\mathrm{\π}/2)=-\sin \mathrm{\θ}\end{array},$ The sine function that namely inputs in CORDIC_POST is the cosine function value of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the sine function of former angle;

If the anglec of rotation that records is ± π, according to the sin cos functions transformation relation $\begin{array}{c}\sin (\mathrm{\θ}\±\mathrm{\π})=-\sin \mathrm{\θ}\\ \cos (\mathrm{\θ}\±\mathrm{\π})=-\cos \mathrm{\θ}\end{array},$ The negative form that namely inputs to the sine function in CORDIC_POST is the sine function of former angle value, and the negative form that inputs to the cosine function value in CORDIC_POST is the cosine function value of former angle;

If need to calculate angle value, according to the angle of rotating that records in CORDIC_PRE, the angle value that is input in CORDIC_POST recovered;

Data-switching after recovering is become accuracy floating-point data and output.

2. a kind of implementation method that covers the single-precision floating point trigonometric function of circle angle as claimed in claim 1, is characterized in that, in step 1, the concrete grammar of the angle that described record rotates is:

Divide as follows angular interval: [π/4, π/4] as the first interval, (π/4,3 π/4] be between Second Region, (3 π/4 ,-3 π/4] be the 3rd interval, (3 π/4 ,-π/4) are the 4th interval

If the input data are the angle value θ of the single-precision floating-point data form of any range, to not belong to [2 π, 2 π] scope in θ be converted to [2 π, 2 π] in scope, be in [2 π for θ, 2 π] scope in situation, only record the angle that the residing angular interval of θ can obtain rotating; If same input data are the vector (x, y) of any range, will (x, y) transform to radius and be in 1 circumference, obtain vector (x ＇, y ＇), only record the angle that (x ＇, y ＇) residing angular interval can obtain rotating;

If angle θ or vector (x ＇, y ＇) are in the first interval, the angle of rotating is zero;

If angle θ or vector (x ＇, y ＇) are between Second Region, the angle of rotating is-pi/2;

If angle θ or vector (x ＇, y ＇) are in the 3rd interval, the angle of rotating is ± π;

If angle θ or vector (x ＇, y ＇) are in the 4th interval, the angle of rotating is+pi/2.

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