CN1137133A - Method and apparatus for quick operation of division - Google Patents
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Abstract
A binary division operation method without divider features use of binary shift and addition to simplify division operation procedure in order to limit error in set range and speed-up operation.
Description
The present invention relates at the shortcoming of long division and the division arithmetic method and apparatus that designs.The present invention can be applicable to the computing of any division, and is suitable especially and in the microprocessor controller of no hardware division instruction, and with raising division arithmetic speed, and the error that makes the merchant is in setting range.
In the past, in the microprocessor controller of no hardware division instruction, generally all adopt long division to make division arithmetic.Two long division computing be a series of subtraction in the hope of quotient, be example with positive integer 8 divided by 3, the calculation step of long division is as follows:
Remove Number and removed Number merchant
－－－－ －－－－ －－－
0,011 1000 0 Bu Sudden 1:0011 1,000 1
－0011
－－－－－－
0101
-0011 Bu Sudden 2:0011------
0010 10
As shown above, dividend 8 is shown 1000 with binary form, and the binary representation of divisor 3 is 0011.In first step, compare the size of divisor and dividend earlier.Because divisor is smaller, earlier dividend is deducted divisor, remainder is 0101.Behind subtraction once, the merchant then advances 1.
In second step, remainder is still greater than divisor.Therefore, again remainder is deducted divisor, obtain 0010; The merchant then advances 1 again, obtains 10.At this moment, remainder is less than divisor, and computing can stop.Last value is 0010, and remainder is 0010.
If do the hand computation long division with subtraction and judgement, its step is as follows:
0010
11_)－－－－
1000
11
－－－－－
10 Bu Sudden 1:
0
11)－－－－
1000
1＜11, the merchant is 0; Bu Sudden 2:
0
11)－－－－
100010＜11, the merchant is 0; Bu Sudden 3:
1
11)－－－－
1000
11
－－－－－
1
100＞11, the merchant is that 1 ， I Number is 1; Bu Sudden 4:
10
11)－－－－
1000
11
－－－－－
10
10＜11，
The merchant is that 10 ， I Number are 10.
The very tediously long complexity of two bit computings and the speed of long division are slow, are not suitable for the simple microprocessor controller of function.For solving the shortcoming of long division, other method is a look-up table.Look-up table must any devision device exclusive disjunction, as long as division table is installed in the read-only memory, is index with divisor and dividend, and the quotient in the retrieval division table gets final product.Though look-up table is more efficient than long division, division table is to be two to the expression mode, and its size is grown up fast, must be taken bigger memory space, is not suitable for the microprocessor of low price.
Based on above-mentioned problem and shortcoming, the present invention mainly is to propose new solution simplifying the division arithmetic program, and saves the memory space that stores division table institute palpus.
Fundamental purpose of the present invention can reach the division arithmetic method and apparatus of division arithmetic effect providing a kind of utilization to table look-up with the skill of displacement and addition (shift=add).
Another object of the present invention is to provide a kind of division arithmetic method and apparatus that must divider, it can simplify the division arithmetic program, and accelerates operation efficiency.
Another purpose of the present invention is providing a kind of division table of saving memory space.
A further object of the present invention is at the division arithmetic method and apparatus that a kind of efficient height and energy departure scope are provided.
Based on above-mentioned purpose, the present invention uses remainder theorem and 1/2 power series polynomial repressentation method to be flow process.Device of the present invention comprises the divisor device that moves to left, and in order to the input divisor, and this divisor is moved to left; The dividend device that moves to left in order to the input dividend, and moves to left this dividend; The controller that moves to left is in order to control the action that this divisor moves to left; Merchant's device that moves to right in order to the retrieval gained of input division table, and moves to right it; Above device utilizes displacement method to reach to take advantage of two or remove two purpose.
The present invention also comprises and door gear and adder.With door gear in order to the move to right output of device of most significant digit output and the merchant of device that moves to left of input dividend, and they are done and (AND) computing.Adder is in order to the input and the output valve of door gear, and will add up with the output valve of door gear.Above device is to utilize addition that the quotient value is added up, and the error that makes last merchant is in predetermined scope.
Store for saving the memory space of division table, division table of the present invention also comprises a unidirectional array quotient value, and the quotient value of this array is a predetermined dividend divided by by 1 operation result to the divisor gained that equals this divisor.
The 1st figure is the conceptual flow chart of division arithmetic of the present invention.
2A figure is a division table of the present invention.
2B figure is the example operation program of the present invention according to the 1st figure.
The 3rd figure is the functional block diagram of most preferred embodiment of the present invention.
The 1st figure is a conceptual flow chart of the present invention.Is example with binary X divided by Y, and Q is scale-of-two merchant's a maximum number of digits, and it requires error limitation at ± (Q+r+1) * 2
-(1+r)Within.The merchant who is deposited in the division table is a division table (i)=2
N+r/ i gets the integer that rounds up.I represents the divisor index, and its size is by 1 to (2
D-1) number, D are represented the maximum number of digits of divisor, and N is the maximum number of digits of dividend, and r is the error reference value.
Its cardinal principle is remainder theorem and 1/2 power time progression polynomial repressentation theorem.Wherein, the function of r for original dividend prior to 2
rStrengthen (2
N+r/ i), to reach the effect that reduces error.And the error range merchant that to be it original is because of 2
rStrengthen event, so be the Q+r position altogether.For asking it to the radix point next bit, so it is inferior to carry out (Q+r+1), and each error is 1/2 to the maximum.So, try to achieve the summation maximum error and be (Q+r+1) * (1/2), at last again with 2
rStrengthen reduction, obtain maximum error to be limited in ± (Q+r+1) * 2
-(1+r)Within.By known conditions N, D, Q, and require error range, can the anti-r of asking for building table.And the size of the division table in the read-only storing apparatus (ROM) is (2
D-1), the figure place of each read-only storing apparatus is the N+r position.
As figure as show, the division arithmetic process flow diagram that the 1st figure is represented, its step is as follows:
100: the result who asks X/Y;
101: whether the value of checking X less than the maximum place value of the dividend that sets, and promptly 2
N, and whether the value of Y subtract 1 value less than the dominant bit of the divisor that sets, promptly 2
(D-1), if satisfy this two conditions, then represent X, the Y value still in the scope that sets, carries out 102; If not, represent that then divisor has reached the maximum divisor index of division table, because of X＞=2
NReached 2
NMaximum dividend scope, and Y＞=2
(D-1), reached the maximum magnitude of division table, if take advantage of 2 (moving to left one) again, then will exceed the division table scope, carry out 103;
102: with X, Y respectively take advantage of 2 effect to reach the molecule denominator, and its value is still constant respectively to moving to left one; Because the divisor dividend respectively increases twice, must get back to 101 to check whether divisor, dividend have reached the scope of setting, and step 101 and 102 is referred to as the displacement approximatioss;
103: the maximum divisor in the Y value representation division table of this moment, tabling look-up obtains discussing reference value, the merchant of the maximum divisor in the division table is temporarily stored among the K K representative merchant device that moves to right;
104: the value among the K is temporarily stored among the T, and it is 0 that number of times is calculated in design, and end value R is 0;
105: whether survey X again less than 2
NThe position, if, then carry out 107, otherwise, carry out 106;
106: the dividend X of this moment has exceeded the maximum dividend 2 that sets
N, therefore X being deducted maximum dividend, the remainder that obtains can continue to do remainder displacement approximatioss; Value among the T and the addition of R value are obtained new end value R;
107:X therefore continue to take advantage of 2 action, and the value time spent among the K moves right less than the maximum dividend that sets, and it act as and removes 2, and afterwards, K value and value of statistical indicant deposit among the T; Value of statistical indicant wherein is as the usefulness that rounds up;
108: calculation times is increased 1;
109: whether calculation times is inferior less than (Q+r+1);
110: with the value among the R and 2
rBe divided by, be the result who is asked;
111: finish.
This result of calculation is for having the binary division result of decimal, if want the round number, need only be by 0 (the casting out) or 1 (carry) of radix point next bit, get final product the result that rounds up, that is immediate integer solution.
For making computing flow process of the present invention more clear, attempt the process flow diagram that an example illustrates the 1st figure.
Please see 2A figure.This figure is a division table of the present invention.Division table of the present invention is the quotient value of an array, and the quotient value of this array is a predetermined dividend divided by by 1 operation result to the divisor gained that equals this dividend.As shown in the figure, its stored numerical value is based on dividend 16, and divisor is by merchant's (value of merchant for rounding up) of 1 to 7 gained.
Suppose that now dividend is 4, divisor is 3, and the merchant is 2, and requiring error is ± 1, asks 10 ÷ 3.By error 1=＜(Q+r+1)/2
(1+r)Can get r=1.Division table (i)=(2
N/+r)/i, i=0-7.
Contrast the process flow diagram of the 1st figure, 2B figure is the operational flowchart of an example.
201: earlier relatively whether dividend 10 greater than 16, and promptly 2
4 Whether divisor 3 less than 4, and promptly 2
3-1.Because divisor and dividend all less than the scope that sets, therefore carry out 202.
202: with 10 and 3 each respectively take advantage of 2 effect to moving to left one to reach the molecule denominator, 201 and 202 employedly take advantage of 2 to move and be referred to as the displacement approximatioss, its purpose is approached set maximum dividend, promptly 16.This be since division table of the present invention only to store dividend be 16, divisor is by 1 to 7 may discuss, if will then dividend must be controlled at more than or equal in 16 the scope, so that isolate 16 part according to look-up table.The dividend of this moment is 20, because of 20＞16, so carry out 203.
203: look into 16/6 merchant, division table (6)=5, its binary representation are 00101.
204: the merchant with gained in 203, promptly 00101 deposit T in, and to set R be 0, R is an end value.
205: dividend 20 is greater than set maximum dividend 16.
206: deduct 6/6 with 20/6, obtain 4/6.R stores the quotient that obtains for the first time, and promptly 5.
207: the molecule of remainder 4/6 is taken advantage of 2.Molecule takes advantage of 2 denominators also must take advantage of 2, and the action that the gained 5 that equals to table look-up is done except that 2 just moves to right one.It is 00101 that K moves to right 00101, and position 1 is sign (flag) one of to move to right out.The result 0010 that T stores after moving to right adds sign 1, and getting the result is 00101, keeps for calculate next time and uses.
208: calculation times is increased 1.
209: survey i and whether reached the number of times that should calculate, because 1＜4, do not reach calculation times, so carry out 210.
210: take out the last reserved value X=8/6 that calculates, K=2, T=3 does not reach 16/6 because of 8＜16, so inferior result is R=5+0=5.
211: the molecule of remainder 8/6 is taken advantage of 2 again, be equivalent to the gained of tabling look-up and remove 2 (moving to right one again) again, K=0010 moves to right and is K=0010, the position that shifted out this moment promptly is masked as 0, T=K+CF=1, and remainder to take advantage of the result after 2 be 16/6, keep for calculate next time and use.
212:i=1+1=2 is not because 2＜4 reach calculation times, so continue execution.
213: take out the last reserved value X=16/6 that calculates, K=1, T=1, dividend 16 calculate this R=R+T=5+1=6 as a result more than or equal to set maximum dividend 16.
214: X=16/6 is deducted 6/6 equal 0/6, take advantage of 2 again, get X=0/6, K moves to right one again, can get K=00001, and T=K+CF=0000+1=0001, K=0000.
215:i=2+1=3, because 3＜4, do not reach calculation times, so continue to carry out.
216: take out the last reserved value of calculating, X=0/6, K=0, T=1, because of dividend 0 less than 16, so inferior R=6+0=6 as a result.
217: with X=0/6 take advantage of 2 X=0/6, the K bit that moves to right again, K=00000, T=K+CF, T=0+0=0 keep for calculate next time and use.
218:i=3+1=4=Q+r+1 has reached calculation times.
219: the result is X/Y=6/2r=3, get error of calculation 10/3-3=1/3 by the result, and (1/3)＜1 is eligible.
The 3rd figure is the functional block diagram of most preferred embodiment of the present invention.As shown in the figure, its device comprises the divisor device 201 that moves to left, the dividend device 202 that moves to left, and the controller 203 that moves to left, division table 204, merchant's device 205 that moves to right is with door gear (AND) 206, adder 207, clock impulse device 208.
As shown in the figure, move to left device 201 of divisor is the D position, stores divisor Y, and the function that moves to left is arranged.Move to left device 202 of dividend is N+1 position (N represents the figure place of dividend).Move to left device 202 of dividend stores dividend X, and the function that moves to left is arranged.Above stream oriented device is moved to the left along with the time clock of clock impulse device 208, and device 201 can be controlled it to the action that moves to left by the controller 203 that moves to left but divisor moves to left.
Above divisor move to left device 201, dividend the move to left device 202 and the controller 203 main displacement approximatiosses of carrying out that move to left.For divisor the move to left most significant digit (MSB) of device 202 of device 201 or dividend that moves to left is 1 o'clock, expression divisor or dividend are greater than default maximum dividend.At this moment, one " 1 " the numerical value input controller 203 that moves to left must be arranged, and import the divisor device 201 that moves to left, make it the action that stops to move to left by the numerical value that control line 212 will " 1 ".Starting signal and start division tables 215 via line 209, is the divisor index value with the divisor after stopping to move to left, and, is tabled look-up divisor index value input division table by line 213.
Division table 215 is contained in the read-only memory 204.The merchant that retrieval obtains imports and discusss in the device 205 that moves to right.Discuss the device 205 that moves to right and carry out the action that moves to right, just carry out effect except that two.When merchant's device 205 that moves to right moves right one the time, dividend moves to left device 202 also to moving to left one, and this is for carrying out the remainder approximatioss.
Input with door gear 206 control adders 207.When dividend moves to right the most significant digit (MSB) of device 202 when being 1, pass through with the value that door gear 206 just allows the merchant move to right in the device 205, and in the input adder 207, added up.The most significant digit (MSB) of device 202 is 0 o'clock if dividend moves to left, and just forbids discussing the value that moves to right in the device 205 with door gear 206 and passes through, and adder 207 is carried out added 0 action.This is for carrying out summation, the merchant of the gained of at every turn tabling look-up is added up, with merchant to the end.
Clock impulse device 208 with the time clock signal be sent to divisor move to left device 201, dividend move to left device 202, merchant move to right device 205, with door gear 206 and adder 207, make it the action that moves to left or move to right along with the time clock signal and carry out the addition action.
Explain with above-mentioned example again, ask 10/3.If N=4, D=3, Q=2, r=1, calculation times is controlled to be Q+r+1=4 time by time clock, and its binary representation is: 1010----, the 0011 divisor number that stores in the device 201 that moves to left is 011.The dividend number that stores in the device 202 that moves to left is 01010.
Because dividend is less than set maximum dividend, the divisor device 201 that moves to left moves to left device 202 simultaneously to moving to left one with dividend, and its number of depositing is respectively 110 and 10100.Dividend 10100 now is 20, greater than 16, so the controller 203 that moves to left makes the divisor device 201 that moves to left stop to be moved to the left, and 0110 to be the divisor index, tables look-up, and obtaining discussing reference value is 00101 (value after rounding up).
Merchant's reference value 00101 deposits in discusss the device 205 that moves to right.The most significant digit unit of device is 1 because dividend moves to left, and therefore, with the input of door gear 206 control adders 207, merchant's reference value 00101 is added in the adder 207.The content that adds adder 207 back adders 207 is 101, and this moment, calculation times was 1, less than 4, continues to carry out.
The dividend device 202 that moves to left continues to be moved to the left according to clock impulse device 208, and its value of depositing is 01000, and the merchant to move to right be 00010, be masked as 1.The most significant digit of device is 0 because dividend moves to left, and therefore, with the input of door gear 206 control adders 207, makes 00010 to be ended with door gear 206, and adder 207 being carried out add behind 0 the result still is 5.This moment, calculation times was 2, less than 4, continued to carry out.
The dividend device 202 that moves to left continues to be moved to the left according to clock impulse device 208 again, and its value of depositing is 10000, and the merchant to move to right be 00001, be masked as 0.The most significant digit of device is 1 because dividend moves to left, and latch means 206 makes the value 00001 of discussing the device 205 that moves to right and indicates that 0 passes through, and by clock impulse device 208 controls, will discuss and indicate input adder 207 respectively, and obtaining the merchant is 110.This moment, calculation times was 3, less than 4, continued to carry out.
The dividend device 202 that moves to left continues again to be moved to the left according to clock impulse device 208, and its value of depositing is 00000, and the merchant to move to right be 0000, be masked as 1.The most significant digit of device is 0 because dividend moves to left, and therefore controls the input of adders 207 with door gear 206, and the value of discussing the device 205 that moves to right is ended, and after adder 207 carried out 0, the result still was 110.This moment, calculation times reached 4 times, and this 110 is the result, and wherein lower r position is a scale, and this example is because of r=1, and so the result is 110, its decade value is 3, and inspection error (10/3)-3=1/3 is less than requiring in error ± 1, so meet the requirements.
The above preferred embodiment only of the present invention, and reached practical effect widely, all equalizations of doing according to claims of the present invention change and modify, and all still belong in the patent coverage of the present invention.
Claims (13)
1. division arithmetic device, this device utilizes displacement and addition to obtain the merchant of divisor and dividend, and this device comprises:
The divisor device that moves to left, in order to the input divisor, and with the lt of this divisor;
The dividend device that moves to left, in order to the input dividend, and with the lt of this dividend;
The controller that moves to left in order to controlling the action that device moves to left that moves to left of this divisor, and stops the action that device moves to left that moves to left of this divisor during greater than a preset value at divisor or dividend;
Merchant's device that moves to right in order to merchant's reference value of input division table retrieval gained, and should be discussed reference value and moved to right;
With door gear, in order to importing this dividend move to left the most significant digit output and this merchant reference value of device, and it is done and action (AND);
Adder will be in order to adding up with the output of door to obtain the result.
2. according to the division arithmetic device of 1 of claim the, the device that moves to left of divisor is wherein exported an index value in order to the index value as the above-mentioned division table of retrieval.
3. according to the division arithmetic device of 1 of claim the, the controller that moves to left wherein also comprises:
Logic sum gate (OR GATE) in order to importing most significant digit output and the above-mentioned dividend of the device most significant digit output of device that moves to left that moves to left of above-mentioned divisor, and carries out exclusive disjunction and exports a controlling signal; And
This controlling signal in order to control above-mentioned divisor move to left device move to left action carrying out with stop, starting the above-mentioned division of retrieval with control.
The 4 division arithmetic devices according to 1 of claim the, division table wherein are the quotient value of an array, and the quotient value of this array is a predetermined dividend divided by by 1 operation result to the divisor gained that equals this dividend.
5. the quick operational method of division utilizes displacement and addition to reach the function of division, and this method comprises the following step:
(1) reads divisor and dividend;
(2) carry out the displacement approximatioss, make this divisor and this dividend, obtain first value more than or equal to a predetermined maximum dividend;
(3) divisor with first value is the indexed search division table, obtains merchant's reference value;
(4) the maximum dividend that will be scheduled to obtains second value divided by the divisor of first value;
(5) this first value is deducted this second value, obtain a remainder;
(6) carry out remainder displacement approximatioss, the molecule that makes this remainder obtains the 3rd value more than or equal to this predetermined maximum dividend;
(7) carry out summation, to get result to the end.
6. according to the division arithmetic method of 5 of claims the, wherein the displacement approximatioss of step (2) comprises step:
(8) when above-mentioned divisor and above-mentioned dividend less than predetermined maximum dividend the time, then above-mentioned divisor and above-mentioned dividend are taken advantage of two simultaneously, obtain the first above-mentioned value.
7. according to the division arithmetic method of 5 of claims the, wherein the displacement approximatioss of step (2) comprises step:
(9) when above-mentioned divisor and above-mentioned dividend during, then stop to take advantage of two action, obtain the first above-mentioned value more than or equal to above-mentioned predetermined maximum dividend.
8. according to the division arithmetic method of 5 of claims the, wherein the remainder displacement of step (6) approaches and comprises step:
(10) when the molecule of above-mentioned remainder less than above-mentioned predetermined maximum dividend, then the molecule of above-mentioned divisor is taken advantage of two action, obtain the 3rd above-mentioned value.
9. according to the division arithmetic method of 5 of claims the, wherein the remainder displacement approximatioss of step (6) comprises step:
(11) when the molecule of above-mentioned remainder takes advantage of two, above-mentioned merchant's reference value removes two simultaneously.
10. according to the division arithmetic method of 5 of claims the, wherein the remainder displacement approximatioss of step (6) comprises step:
(12) the specification error scope is at (Q+r+1) * 2
-(1+r)Within.
11. according to the division arithmetic method of 5 of claims the, wherein the summation of step (7) comprises step:
(13) set-up and calculated reference value is as the number of times that adds up.
12. according to the division arithmetic method of 5 of claims the, wherein the summation of step (7) comprises step:
(14) with above-mentioned merchant's reference value and the 3rd above-mentioned value addition.
13. according to the division arithmetic method of 5 of claims the, wherein the summation of step (7) comprises step:
(15) when the molecule of above-mentioned the 3rd value more than or equal to above-mentioned predetermined maximum dividend, execution in step (7).
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN 95107302 CN1137133A (en) | 1995-05-30 | 1995-05-30 | Method and apparatus for quick operation of division |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN 95107302 CN1137133A (en) | 1995-05-30 | 1995-05-30 | Method and apparatus for quick operation of division |
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| Publication Number | Publication Date |
|---|---|
| CN1137133A true CN1137133A (en) | 1996-12-04 |
Family
ID=5076257
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| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN 95107302 Pending CN1137133A (en) | 1995-05-30 | 1995-05-30 | Method and apparatus for quick operation of division |
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Cited By (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN100334543C (en) * | 2003-12-16 | 2007-08-29 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100334542C (en) * | 2003-12-16 | 2007-08-29 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100342323C (en) * | 2003-12-16 | 2007-10-10 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100349110C (en) * | 2003-12-16 | 2007-11-14 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353312C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353310C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353311C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN102063284A (en) * | 2010-12-13 | 2011-05-18 | 青岛海信信芯科技有限公司 | Division operation method and device |
| CN101685384B (en) * | 2008-09-23 | 2011-10-26 | 华晶科技股份有限公司 | Integer division operational circuit tolerating errors |
-
1995
- 1995-05-30 CN CN 95107302 patent/CN1137133A/en active Pending
Cited By (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN100334543C (en) * | 2003-12-16 | 2007-08-29 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100334542C (en) * | 2003-12-16 | 2007-08-29 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100342323C (en) * | 2003-12-16 | 2007-10-10 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100349110C (en) * | 2003-12-16 | 2007-11-14 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353312C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353310C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN100353311C (en) * | 2003-12-16 | 2007-12-05 | 河北工业大学 | A fast divider with divisor 15x2n |
| CN101685384B (en) * | 2008-09-23 | 2011-10-26 | 华晶科技股份有限公司 | Integer division operational circuit tolerating errors |
| CN102063284A (en) * | 2010-12-13 | 2011-05-18 | 青岛海信信芯科技有限公司 | Division operation method and device |
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