CN1472635A - Floating number index number computing method and device - Google Patents

Abstract

The device is used to obtain an index operation result of floating point with Z as of radix and it includes a converting device to receive floating point input and to output an integer and a fraction part, K index league table to receive one share of N number bit elements as N=N1+N2+...+Nk and to generate an output result by checking the table, a multiplier to receive an output result input of index league table and to produce a mentissa, of which the integer part is an index and a index operation result is represented with mentissa, index and an output symbol number with value of zero in formula of (-1)sy.2Ey.my as sy referring to symbol number, Ey to index, my to mentissa as k<=my<2, N,K,K1,K2...Kn to natural number.

Description

The index operation method of floating number and device

Technical field

The invention relates to a kind of method and apparatus of exponent arithmetic, particularly about a kind of index operation method and device of floating number.

Background technology

In present robot calculator, the most frequently used representation of floating number F is:

F＝M×β ^E

Wherein M is mantissa (mantissa), and E is an index, and β is the radix of index.

(the Instoyute of Electrical and ElectronicEngineers of motor and Electronic Engineering Association, IEEE) four kinds of standard formats have been concluded for the representation of floating number, preceding two kinds of forms are single accurate 32 bit forms (single-precision 32-bit format) and the accurate 64 bit forms of dibit (double-precision 64-bit format), other two kinds of intermediate results when extending form and be used to represent computing.For single accurate 32 bit format notations, most important purpose promptly is the degree of accuracy of performance floating number, and only when obtaining more number of significant digit, just utilize the accurate 64 bit format notations of dibit to use Double Length (Double Length) storage area to deposit this floating number.

Consult Fig. 1, Fig. 1 shows the synoptic diagram of above-mentioned single accurate 32 bit format notations.In this representation, 2 being radix, floating number F=(1) ^S2 ^EM, wherein M is the mantissa (mantissa) of this floating number, uses 23 bits to represent, and E is the index of this floating number, uses 8 bits to represent, and the symbolic number that S counts for this symbol uses 1 bit to represent.

In present robot calculator, the computing of floating number is nearly all adopted in all computings, so the operation efficiency of floating number has determined the usefulness of this robot calculator.And in present way, usually all use the method for tabling look-up, set up comparison list in advance, when computing, cooperate and table look-up, in the hope of the exponent arithmetic result of floating number, and when doing the exponent arithmetic of floating number with the method, can meet the problem of some computing degree of accuracy, the table of comparisons of one 8 bit has been quite huge, but uses the table of comparisons of 8 bits to do the exponent arithmetic of floating number, the degree of accuracy of its operation result is still not enough, because the part of a common floating number mantissa all has 23 bits.

Summary of the invention

In view of this, fundamental purpose of the present invention is to provide a kind of exponent arithmetic apparatus and method of floating number, is used to obtain have the finger counting operation result of the floating number of pinpoint accuracy.

For reaching above-mentioned purpose, the invention provides a kind of exponent arithmetic device of floating number, being used to obtain a floating number is an exponent arithmetic result at the end with 2, the representation of this floating number is (1) ^Sx2 ^ExM _x, this exponent arithmetic result's representation is (1) ^Sy2 ^EyM _y, wherein Sx is that symbolic number, the Sy of this floating number are this exponent arithmetic result's index, m for this exponent arithmetic result's symbolic number, Ex for index, the Ey of this floating number _xMantissa, m for this floating number _yMantissa and 1≤m for this exponent arithmetic result _x＜2,1≤m _y＜2, this exponent arithmetic device comprises: a conversion equipment in order to the index of the symbolic number that receives above-mentioned floating number, above-mentioned floating number and mantissa's input of above-mentioned floating number, converts above-mentioned floating number to an integral part and fractional part output; K the index table of comparisons, above-mentioned fractional part have N bit, and N bit is divided into K part, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, each index table receives the portion input in the above-mentioned K part, and the generation one output result that tables look-up; One multiplier is imported in order to the output result who receives above-mentioned each index table of comparisons, and produces above-mentioned exponent arithmetic result's mantissa.The integral part of wherein above-mentioned conversion equipment output is above-mentioned exponent arithmetic result's a index, and its value of above-mentioned exponent arithmetic result's symbolic number is zero, and above-mentioned N, K, K ₁, K ₂..., K _NBe natural number.

On the other hand, the present invention also provides a kind of index operation method of floating number, and being used to obtain a floating number is an exponent arithmetic result at the end with 2, and the representation of this floating number is (1) ^Sx2 ^ExM _x, wherein Sx is that symbolic number, the Ex of this floating number are index, the m of this floating number _xBe the mantissa of this floating number, and 1≤m _x＜2, this index operation method comprises the following steps: at first, and the index of the symbolic number of above-mentioned floating number, above-mentioned floating number and the mantissa of above-mentioned floating number are imported in the conversion equipment; Then, in above-mentioned conversion equipment, above-mentioned floating number is converted to an integral part and a fractional part; Above-mentioned fractional part is divided into K part, and above-mentioned fractional part has N bit, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, each index table receives the portion input in the above-mentioned K part, and the generation one output result that tables look-up; Then, the output result with above-mentioned each index table imports generation one mantissa in the multiplier; Utilize above-mentioned mantissa, above-mentioned integral part at last, and a value is that zero symbolic number is represented above-mentioned exponent arithmetic result, this exponent arithmetic result's representation is (1) ^Sy2 ^EyM _y, wherein Sy is that this symbolic number, Ey are above-mentioned integral part, m _yBe above-mentioned output mantissa, and 1≤m _y＜2, and above-mentioned N, K, K ₁, K ₂..., K _NBe natural number.

Description of drawings

Fig. 1 represents the synoptic diagram of known single accurate 32 bit format notations;

Fig. 2 represents the configuration diagram of the exponent arithmetic device of floating number of the present invention;

Fig. 3 represents the configuration diagram of conversion equipment of the present invention;

Fig. 4 represents the configuration diagram of exponent arithmetic device of the floating number of the embodiment of the invention.The figure number explanation:

M, m _xThe mantissa of-floating number;

The index of E, Ex-floating number;

The symbolic number that S, Sx-symbol is counted;

The Ix-integral part;

The Fx-fractional part;

The 10-conversion equipment;

The 12-shift unit;

The 14-pick-up unit;

The 16-determination device;

20 ₁-20 _K-index the table of comparisons;

The 30-multiplier;

The Fsc-fractional part that is shifted;

The output of Isc-displacement integral part;

The Err-error messages;

Sy-exponent arithmetic result's symbolic number;

Ey-should count the index of operation result;

m _y-exponent arithmetic result's mantissa.

Embodiment

Fig. 2 represents the configuration diagram of the exponent arithmetic device of floating number of the present invention.As shown in the figure, the exponent arithmetic device of floating number comprises conversion equipment 10, a K index table of comparisons 20 ₁-20 _KAnd multiplier 30.Conversion equipment 10 is in order to the index E x of the symbolic number Sx, this floating number X that receive a floating number X and the m of mantissa of this floating number X _xInput converts this floating number to an integral part Ix and fractional part Fx output and output error message Err in the time this floating number can't being expressed as integral part and fractional part.Fractional part Fx has N bit, and N bit is divided into K part, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, K the index table of comparisons 20 ₁-20 _KIn each index table receive portion input in the above-mentioned K part, and table look-up and produce an output result.Multiplier 30 is imported in order to the output result who receives each index table, and produces the exponent arithmetic m of mantissa of Y as a result _yWherein the integral part Ix of conversion equipment 10 output be this exponent arithmetic as a result Y index E y and because exponent arithmetic as a result Y all be positive number, so exponent arithmetic as a result the symbolic number Sy of Y be zero.

The representation of above-mentioned floating number X is:

X＝(-1) ^Sx·2 ^Ex·m _x (1)

Wherein Sx is the symbolic number of this floating number, and when floating number X was positive number, the symbolic number Sx of floating number was 0.When floating number X was negative, the symbolic number Sx of floating number was 1; Ex is the index of this floating number; m _xBe the mantissa of this floating number, and 1≤m _x＜2.

The exponent arithmetic device of floating number of the present invention, being used to obtain this floating number X is the exponent arithmetic Y as a result at the end with 2:

Y＝2 ^X＝(-1) ^Sy·2 ^Ey·m _y (2)

Wherein Sy is this exponent arithmetic result's a symbolic number because exponent arithmetic as a result Y all be positive number, so exponent arithmetic as a result the symbolic number Sy of Y be zero; Ey is this exponent arithmetic result's a index; m _yBe this exponent arithmetic result's mantissa, and 1≤m _y＜2.

In order to try to achieve Y, method of the present invention is divided into integral part and fractional part with X earlier:

X＝(-1) ^Sx·2 ^Ey·m _x＝Ix+Fx (3)

Wherein I x is that integral part, Fx are fractional part and 0≤Fx＜1.

Fx＝q·2 ^-N＝(Ai·2 ^Ni)·2 ^-N (4)

Wherein q is a N bit numeral, and Ai is a Ni bit numeral. $Y=2^{\mathrm{Ix}+\mathrm{Fx}}=2^{\mathrm{Ix}}\&times;\left[\underset{i}{\mathrm{\&Pi;}}{2}^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}}\right]----\left(5\right)$

Therefore, exponent arithmetic result's index E y, this exponent arithmetic result's the m of mantissa _yAnd exponent arithmetic as a result the symbolic number Sy of Y be respectively:

Ey＝Ix (6) $\mathrm{my}=2\mathrm{Fx}=\underset{i}{\mathrm{\&Pi;}}{2}^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}}=\underset{i}{\mathrm{\&Pi;}}\mathrm{Ti}----\left(7\right)$

Sy＝0 (8)

And in $\mathrm{Ti}=\underset{i}{\mathrm{\&Pi;}}{2}^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}},$ And 1≤i≤K, K are the number of the index table of comparisons, and because 0≤Fx＜1, so 2 ⁰≤ 2 ^Fx＜2 ¹, promptly 1≤2 ^Fx＜2, therefore satisfy 1≤m _y＜2.

Utilizing the exponent arithmetic device of floating number of the present invention can obtain this floating number X is the exponent arithmetic Y as a result at the end with 2, at first, and with the index E x of symbolic number Sx, the floating number X of floating number X and the m of mantissa of floating number X _xIn the input conversion apparatus 10.Then, in conversion equipment 10, floating number X is expressed as integral part Ix and fractional part Fx (with reference to the 3rd formula).Then, above-mentioned fractional part is divided into K part, above-mentioned fractional part has N bit, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, each index table receives the portion input in the above-mentioned N part, and tables look-up and produce an output result, the output result of each index table is imported produce the exponent arithmetic m of mantissa of Y as a result in the multiplier 30 again _y(with reference to the 4th, 5 and 7 formulas).At last, utilize the m of mantissa _y, integral part Ix, and a value is zero symbolic number Sy, the expression exponent arithmetic is Y as a result, this exponent arithmetic representation of Y as a result is (1) ^Sy2 ^EyM _y, wherein Sy is that its value of symbolic number is zero (with reference to the 8th formula), Ey is integral part Ix (with reference to the 6th formula), m _yBe mantissa and 1≤m _y＜2.

Fig. 3 represents the configuration diagram of conversion equipment of the present invention.As shown in the figure, this conversion equipment 10 comprises shift unit 12, pick-up unit 14 and determination device 16.Shift unit 12 is in order to the index E x of reception floating number and the m of mantissa of floating number _xInput is according to the index E x of the floating number m of mantissa with floating number _xDisplacement, for instance, when the index E x of floating number positive integer, then according to the mantissa m of this positive integer with floating number _xTo moving to left, for example: the index E x of floating number is 5, then the m of mantissa of floating number _xTo 5 bits that move to left, when the index E x of floating number negative integer, then according to the mantissa m of this positive integer with floating number _xMove right, for example: the index E x of floating number is-1, then the m of mantissa of floating number _x1 bit that moves right, and shift unit 12 produces a displacement fractional part Fsc and a displacement integral part Isc output.

Pick-up unit 14 sends error messages Err in order to detect shift unit 12 when shift unit 12 overflows.Determination device 16 is in order to receive the symbolic number Sx input of displacement integral part Isc and floating number, and the symbolic number Sx according to floating number determines the sign of displacement integral part Isc to export this conversion equipment 10 to produce integral part Ix, when Sx is 1, Ix=-Isc, when Sx is 0, Ix=Isc.The fractional part that wherein is shifted Fsc is fractional part Fx after exporting this conversion equipment 10.

Fig. 4 represents the configuration diagram of exponent arithmetic device of the floating number of the embodiment of the invention.As shown in the figure, the exponent arithmetic device of floating number comprises shift unit 12, pick-up unit 14,16,3 index tables of comparisons 20 of determination device ₁-20 ₃And multiplier 30.The exponent arithmetic device of the floating number of present embodiment be used to obtain this floating number X with 2 be the end exponent arithmetic as a result Y be Y=2 ^X, the representation of this floating number X is X=(1) ^Sx2 ^ExM _x, wherein Sx is the symbolic number of this floating number, represents with 1 bit, and when floating number X was positive number, the symbolic number Sx of floating number was 0, and when floating number X was negative, the symbolic number Sx of floating number was 1; Ex is the index of this floating number, represents with 8 bits; m _xBe the mantissa of this floating number, represent with 24 bits, and 1≤m _x＜2.

In order to try to achieve Y, need earlier X to be divided into the integral part Ix that represents with 8 bits and the fractional part Fx that represents with 23 bits, wherein 0≤Fx＜1; Shift unit 12 receives the index E x of floating number and the m of mantissa of floating number _xInput is according to the index E x of the floating number m of mantissa with floating number _xDisplacement, for instance, when the index E x of floating number positive integer, then according to the mantissa m of this positive integer with floating number _xTo moving to left, for example: the index E x of floating number is 5, then the m of mantissa of floating number _xTo 5 bits that move to left, when the index E x of floating number negative integer, then according to the mantissa m of this positive integer with floating number _xMove right, for example: the index E x of floating number is-1, then the m of mantissa of floating number _x1 bit that moves right, and shift unit 12 produces a displacement fractional part Fsc and a displacement integral part Isc output.Pick-up unit 14 sends error messages Err in order to detect shift unit 12 when shift unit 12 overflows, this error messages Err comprises the overflow message and to the underflow bit message.When the index E x of floating number greater than 7 the time, the m of mantissa of floating number _xSurpass 7 bits to moving to left, promptly send the overflow message this moment.When the index E x of floating number less than-23 the time, the m of mantissa of floating number _xMoving right surpasses 23 bits, and promptly send to the underflow bit message this moment.Determination device 16 is in order to receive the symbolic number Sx input of displacement integral part Isc and floating number, and the symbolic number Sx according to floating number determines the sign of displacement integral part Isc to export this conversion equipment 10 to produce the integral part Ix that represents with 8 bits, when Sx is 1, Ix=-Isc, when Sx is 0, Ix=Isc.

The fractional part that wherein is shifted Fsc is the fractional part Fx integral part of utilizing 23 bits to represent behind this conversion equipment 10 of output, and integral part Ix is the exponent arithmetic index E y of Y (with reference to the 6th formula) as a result.

Then, fractional part Fx is divided into 3 parts, be respectively 8 bits, 8 bits and 7 bits, in regular turn with in each part input index table of comparisons 20,22,24, and in each index table of comparisons, produce an output result, the output result of each index table is imported produce the exponent arithmetic m of mantissa of Y as a result in the multiplier 30 again _y

According to the 7th formula: $m_y=2^{\mathrm{Fx}}=\underset{i}{\mathrm{\&Pi;}}{2}^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}}=\underset{i}{\mathrm{\&Pi;}}\mathrm{Ti}$

In this embodiment, 1≤i≤3,3 are the number of the index table of comparisons, so: $m_y=\mathrm{\&Pi;Ti}={\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="A0212709100141.png,A0212709100151.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\&times;T_2\&times;T_3=2^{A_1\&times;2^{-8}}\&times;2^{A_2\&times;2^{-16}}\&times;2^{A_3{\&times;2}^{-23}}----\left(9\right)$

Wherein ${\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="A0212709100141.png,A0212709100151.png"\; state="\{\{state\}\}">T</figure-callout>}}_1=2^{A_1\&times;2^{-8}},$ And A ₁First in 3 parts that are divided into for fractional part Fx is the numeral of one 8 bits; $T_2=2^{A_2\&times;2^{-16}},$ And A ₂Second portion in 3 parts that are divided into for fractional part Fx is the numeral of one 8 bits; $T_3=2^{A_3\&times;2^{-23}},$ And A ₃Last part in 3 parts that are divided into for fractional part Fx is the numeral of one 7 bits.

Because exponent arithmetic Y as a result all is a positive number, thus exponent arithmetic as a result the symbolic number Sy of Y be zero (with reference to the 8th formula).

At last, utilize m _y, Ix, and Sy represents exponent arithmetic Y as a result, its representation is Y=(1) ^Sy2 ^EyM _y, wherein Sy is that this exponent arithmetic result's symbolic number, Ey are this exponent arithmetic result's index, m _yBe this exponent arithmetic result's mantissa, and 1≤m _y＜2.

Claims (12)

1. the exponent arithmetic device of a floating number, being used to obtain a floating number is an exponent arithmetic result at the end with 2, the representation of this floating number is (1) ^Sx2 ^ExM _x, this exponent arithmetic result's representation is (1) ^Sy2 ^EyM _y, wherein Sx is that symbolic number, the Sy of this floating number are this exponent arithmetic result's index, m for this exponent arithmetic result's symbolic number, Ex for index, the Ey of this floating number _xMantissa, m for this floating number _yBe this exponent arithmetic result's mantissa, and 1≤m _x＜2,1≤m _y＜2, this exponent arithmetic device comprises:

One conversion equipment in order to the index of the symbolic number that receives above-mentioned floating number, above-mentioned floating number and mantissa's input of above-mentioned floating number, converts above-mentioned floating number to an integral part and fractional part output;

K the index table of comparisons, above-mentioned fractional part have N bit, and N bit is divided into K part, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, each index table receives the portion input in the above-mentioned K part, and the generation one output result that tables look-up; And

One multiplier is imported in order to the output result who receives above-mentioned each index table of comparisons, and produces above-mentioned exponent arithmetic result's mantissa;

The integral part of wherein above-mentioned conversion equipment output is above-mentioned exponent arithmetic result's a index, and its value of above-mentioned exponent arithmetic result's symbolic number is zero, and above-mentioned N, K, K ₁, K ₂..., K _NBe natural number.

2. the exponent arithmetic device of floating number according to claim 1, it is characterized in that: this conversion equipment comprises:

One shift unit in order to mantissa's input of the index that receives above-mentioned floating number and above-mentioned floating number, according to the mantissa's displacement with above-mentioned floating number of the index of above-mentioned floating number, and produces a displacement fractional part and a displacement integral part output; And

One determination device is imported in order to the symbolic number that receives above-mentioned displacement integral part and above-mentioned floating number, and is determined the sign of above-mentioned displacement integral part to produce above-mentioned integral part according to the symbolic number of above-mentioned floating number.

3. the exponent arithmetic device of floating number according to claim 2 is characterized in that: more comprise a pick-up unit, in order to detecting above-mentioned shift unit, send an error messages when above-mentioned shift unit overflow.

4. the exponent arithmetic device of floating number according to claim 2, it is characterized in that: this displacement fractional part is above-mentioned fractional part

5. the exponent arithmetic device of floating number according to claim 1, it is characterized in that: above-mentioned i index table receives the input of the i part A i in the above-mentioned fractional part, and above-mentioned fractional part has the N bit, and Ai has the Ni bit, and its output Ti as a result is: $\mathrm{Ti}=2^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}},$ Wherein i is a natural number.

6. the exponent arithmetic device of floating number according to claim 1, it is characterized in that: when above-mentioned floating number was positive number, the symbolic number of above-mentioned floating number was 0, when above-mentioned floating number was negative, the symbolic number of above-mentioned floating number was 1.

7. the index operation method of a floating number, being used to obtain a floating number is an exponent arithmetic result at the end with 2, the representation of this floating number is (1) ^Sx2 ^ExM _x, wherein Sx is that symbolic number, the Ex of this floating number are index, the m of this floating number _xBe the mantissa of this floating number, and 1≤m _x＜2, this index operation method comprises the following steps:

The index of the symbolic number of above-mentioned floating number, above-mentioned floating number and the mantissa of above-mentioned floating number are imported in the conversion equipment;

In above-mentioned conversion equipment, above-mentioned floating number is converted to an integral part and a fractional part;

Above-mentioned fractional part is divided into K part, and above-mentioned fractional part has N bit, and each part has N respectively ₁, N ₂..., N _KBit, and N=N ₁+ N ₂+ ...+N _K, each index table receives the portion input in the above-mentioned K part, and the generation one output result that tables look-up;

The output result of above-mentioned each index table is imported generation one mantissa in the multiplier; And

Utilize above-mentioned mantissa, above-mentioned integral part, and a value is that zero symbolic number is represented above-mentioned exponent arithmetic result, this exponent arithmetic result's representation is (1) ^Sy2 ^EyM _y, wherein Sy is that this symbolic number, Ey are above-mentioned integral part, m _yBe above-mentioned output mantissa, 1≤m _y＜2, and above-mentioned N, K, K ₁, K ₂..., K _NBe natural number.

8. the index operation method of floating number according to claim 7 is characterized in that: the step that in above-mentioned conversion equipment above-mentioned floating number is expressed as an integral part and a fractional part comprises following substep:

Utilize a shift unit, receive the index of above-mentioned floating number and mantissa's input of above-mentioned floating number, the index of the above-mentioned floating number of foundation is shifted the mantissa of above-mentioned floating number, and produces a displacement fractional part and a displacement integral part output; And

Utilize a determination device, receive the symbolic number input of above-mentioned displacement integral part and above-mentioned floating number, and determine the sign of above-mentioned displacement integral part to produce above-mentioned integral part according to the symbolic number of above-mentioned floating number.

9. the index operation method of floating number according to claim 8 is characterized in that: the step that in above-mentioned conversion equipment above-mentioned floating number is expressed as an integral part and a fractional part more comprises substep:

Utilize a pick-up unit, when above-mentioned shift unit overflow, send above-mentioned error messages.

10. the index operation method of floating number according to claim 8, it is characterized in that: this displacement fractional part is above-mentioned fractional part.

11. the index operation method of floating number according to claim 7 is characterized in that: above-mentioned i index table receives the input of the i part A i in the above-mentioned fractional part, and above-mentioned fractional part has the N bit, and Ai has the Ni bit, and its output Ti as a result is: $\mathrm{Ti}=2^{{\mathrm{Ai}\&times;2}^{\mathrm{Ni}-N}},$ Wherein i is a natural number.

12. the index operation method of floating number according to claim 7 is characterized in that: when above-mentioned floating number was positive number, the symbolic number of above-mentioned floating number was 0, and when above-mentioned floating number was negative, the symbolic number of above-mentioned floating number was 1.

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