JPH05508950A - Apparatus and method for estimating logarithms - Google Patents

Abstract

(57) [Summary] This bulletin contains application data before electronic filing, so abstract data is not recorded.

Description

[Detailed description of the invention] Apparatus and method for estimating logarithms field of invention The present invention relates to transcendental function estimation (transcendental function estimation). stimulation: TEF). More specifically, the present invention provides a logarithmic function The present invention relates to a method and apparatus for efficiently estimating numbers.

Background of the invention More efficient calculation times to improve the speed of logarithmic operations in computers avenues and methods are needed.

Typically, this logarithmic system utilizes a hardware fixed-point processor. , this processor is A Unified Algorithm For Ele Mentary Functions, SpringJoint Comput ter Conference, +971 W a I (hCl C0RDIC (Coordina te Rotation Digital Computer) calculation method, Al in the description of Tien ChiChen of National Patent No. 3,631,230 It has a algorithm. Both C0RDIC and Chen's systems are designed for efficiency. requires hardware configuration. Save hardware costs and reduce physical This is because it provides an alternative way to improve space conditions.

Summary of the invention A method and apparatus for substantially processing a human input value to provide an output logarithm value of this input value is provided. is provided, and this output log value has the desired base. modifier ) uses this input value to generate an approximate value, and the first function generator uses this approximate value. Then, a first intermediate value is determined. This first intermediate value is essentially 1) of the output logarithm value. If the desired base is e, the preselected natural base logarithm (natural b approximation obtained directly from the first function generator that generates 2) if the desired base of the logarithm value of the first function generator is not e. the logarithm of the approximation found to the desired base of the first function generator and the first function generator. It is the product of the natural base logarithm of the desired base of the natural instrument.

The error generator uses the input value and the approximate value to , gives the error value. The correction evaluator is A correction value is provided using this error value and a predetermined number of terms of a specific power series. synthesis 1) When the intermediate value is a natural base logarithm, the combination is the intermediate value and the correction value. and 2) the scale of the intermediate value and the correction value. (However, scaling is the natural base of the desired base of the output logarithm value.) Either dividing both terms individually or combining these two terms by the logarithm (consisting of dividing the sum) and calculate the resulting output logarithm value.

Brief description of the drawing FIG. 1A is a block diagram of the computer hardware configuration of the present invention. Figure B shows one embodiment of the first function generator of Figure 1A, and Figure 1C shows one embodiment of the first function generator of Figure 1A. One embodiment of the error generator is shown, and FIG. 1D is one embodiment of the synthesizer of FIG. 1A. shows.

FIG. 2A is a general flowchart for carrying out the method of the present invention), and FIG. 2B is a general flowchart for carrying out the method of the present invention. One embodiment of the steps for calculating the first intermediate value is shown in FIG. 2C, where log, , ( One configuration of the step of determining x) is shown.

FIG. 3 shows a diagram of a diagram for implementing the method of the invention which gives the output logarithmic value of the input value to the base m. It is a low chart.

BEST MODE FOR CARRYING OUT THE INVENTION FIG. 1A, generally designated by the reference numeral 100, depicts a computer hardware system of the present invention. An example of a hardware configuration is shown below.

A modifier (102) uses the input value to determine an approximate value. , this approximation value is chosen from a predetermined set of values. This predetermined set of values is rounding algorithms A collection of approximations that are less than or equal to the bit-precision number of the input value. gives the approximate value as a value with a precision value.

a first function generator (106), an embodiment of which is shown in FIG. 1B (reference numeral 125);

) determines the first intermediate value using an approximate value, and this first intermediate value is substantially the approximate value 0 general, which is the natural base logarithm of The first function generator (106) includes at least a ROM, a lookup table or or another memory device as the second function generator (101), and has a predetermined base. Determine the first logarithm value, and use the third function generator (103) to calculate the first logarithm value. Determine the second natural base logarithmic value of a given base of the value, and use a first sealer ( 105), the product of the first log value and the second log value (effectively the first intermediate value) is decide. If the first logarithm has a base C, then the logarithm of the given base of the first logarithm is This can be omitted. Because the value of its logarithm is simply a scale multiplier factor of 1 ( scaling mul plicative factor), and this This is because the 1-log value is substantially the first intermediate value.

Generally, in a software configuration, the first function generator generates an approximation value in at least one construct. at least one first integer for the memory device, scaled by a constant that depends on the configuration. @Memory device that has an integral valued 1ndex By using this at least one first integer exponent, A numerical value and a second natural base logarithm value are determined.

Various memory devices are available to store the first and second log values. This is clear to those skilled in the art.

A common hardware configuration directly utilizes the approximate bit pattern to reduce You can store at least one second integer exponent value, and then use this value to calculate the first pair. A numerical value and a second logarithm value can be included. The first adjustment determined by the software configuration Note that the numerical exponent may be different from the #2 integer exponent generated by the hardware configuration. is clear to a person skilled in the art.

The error generator (104), an embodiment of which is shown in FIG. 1C (reference numeral 150), is , calculate the first quotient by dividing the input value by the approximate value using the first divider (107), and By calculating the difference between the first quotient and 1 using the subtracter (109), the error value is effectively calculated. Hit the target. Equivalently, this subtracter can be used to calculate the difference between the input value and the approximate value. and the related linear value obtained by dividing this difference by an approximation using the first divider. It is clear that the primary quotient is important.

A correction evaluator (108) A determiner that utilizes a difference value and a predetermined number of terms in a predetermined mathematical series. er), which effectively determines the correction value by constructing a predetermined number of terms. and substantially add the estimated predetermined number of terms to substantially correct it. Find a second sum equal to the value. In general, a given mathematical series is a power series of : However, C is an error value. That is, cut a given mathematical series down to a given number of terms. The correction value is effectively found by discarding and estimating the following term: However, C is an error value, and SV is a numerical value of a predetermined number of terms.

The numerical value (SV) of a given number of terms of a given mathematical series is generally an integer value; The value (SV) is effectively the selected value, which is generally the output value. The desired bit precision value of the force logarithm value divided by the bit precision value of the first function generator. It is greater than or equal to the value obtained by subtracting the 3 quotient from 1. Regarding the output logarithm To achieve the desired accuracy, this selected value is used as described above. must be at least equal to 1 minus the third quotient.

combiner (110) (this embodiment is shown in FIG. 1D (reference number 175). ) calculates the output logarithm value using the first intermediate value and the correction value. one Generally, the synthesizer performs 1) scaling operations on the correction value and the first intermediate value. ingoperation) and add both scaled values, or Or, 2) Equivalently, add the correction value to the first intermediate value using the second adder (111). to calculate the second sum, and use the fourth function generator (113) to calculate the third logarithm value (this value is generally the natural base logarithm of the desired base of the output log value), and a second divider Using (115), we can reduce this second sum by dividing it by the third logarithm. Measure. If the desired base of the output logarithm is C, the scaling operation is division by 1, omitted It is clear that it can be done.

FIG. 2A is a schematic flowchart showing the configuration of the method of the present invention, and FIG. The figures and FIG. 2C illustrate certain steps as follows. Sunawa FIG. 2A, generally designated by the reference numeral 200, shows the method of the present invention carried out in the method of the present invention. 206 shows one embodiment of a series of steps to Figure 2B shows an embodiment of one configuration of the step of determining the first intermediate value, generally designated by reference number 7. Figure 2C, denoted by number 212, shows the implementation of one configuration of the step of calculating the log equation (X). Give an example.

FIG. 2A shows one component of the method of the invention. Input value X (202) is modified , obtain an approximation value a = mad(x) represented by at least one electrical signal. (204). Generally, this approximation value is selected from a predetermined set of values, and this predetermined The set of values is a set of values predetermined by the selected rounding algorithm. It is. Therefore, the approximation is based on the particular chosen rounding algorithm, substantially have a bit precision value less than or equal to the bit precision value of .

The approximate value is used to calculate a first intermediate value (206).

A typical first intermediate value (FIV) determination is shown in Figure 2B (reference numeral 225); Here, using a memory device (MEM device) and a, the first logarithm value has a predetermined base p. and the first of the approximations, as represented by at least one electrical signal. Calculate the logarithm value (207). Furthermore, the third natural logarithm value of the predetermined base p is determined (20 9), which is represented by at least one electrical signal. FIV is the first logarithm It is essentially calculated as the product of the value and the third logarithm value (211). given the first logarithm value If the base p is e, the determination of the first logarithm value yields the FIV, which effectively The step of determining the third logarithm value which is 1 can be omitted, and the third logarithm value and the first logarithm The step of determining the product with the value (this product is equal to the first logarithm value) can be omitted. is clear. Also, since FIV is essentially the logarithm of the approximate value, its precision ( It is clear that the precision representing the logarithm of the input value) is smaller than the precision of the input value. death However, the precision of FIV (the precision of calculating the logarithm) is at least the desired precision of the output logarithm. must be equal.

at least the input value and the approximate value are represented by at least one electrical signal Used to determine the error value (20g). In general, divide the input value by an approximation The first quotient is determined, and this first quotient is divided by 1 to essentially obtain the error value. Ru. Other mathematical operations equivalent to the error value determination described above, e.g. subtracting the approximate value from the input value Find the first difference by It can be used in place of the top.

at least the error value determines a correction value represented by at least one electrical signal (210).

Generally, error values are entered into a given number of terms in a given mathematical series, and these are added together. to substantially obtain the correction value. The number of terms for a given number is selectable, and this selection Depends on the desired precision of the logarithmic value. In general, it is possible to obtain the desired precision of the output logarithm value by To give a possible correction value, the number of terms given the desired bits of the output logarithm value greater than 1 minus the quotient of the precision number effectively divided by the number of bits in the FIV or is chosen to be equal. In general, a given mathematical series is a power series This power series is the Tiller series for the natural logarithm of the first sum of l and the error value. (Taylor 5eries) and is substantially represented by the following formula: However, C is an error value.

As shown in Figure 2A, the input value can be output using at least the FIV and the correction value. A force logarithm value is substantially obtained, where the output logarithm has the desired base and is at least one (212). That is, FIG. 2C (reference number 25 0), generally a second sum of the correction value and the FIV is obtained (213) and the input The fourth natural logarithm of the desired base of the output logarithm of the force value is obtained (215), and the second sum is divided by the fourth logarithm, and this quotient is effectively the output logarithm of the input value. , which has the desired bottom and is represented by at least one electrical signal (21 7).

The predetermined base p of the first logarithm value is substantially the natural base C, and the output logarithm value of the input value If the desired base of is also substantially the natural base C, the step of finding the FIV (206) is , it is time to calculate the natural base logarithm of the approximate value, and this natural base logarithm is essentially FIV. Ru. Similarly, multiply the output log value of the input value such that the output log value has the desired base. The step is to calculate the second sum of the FIV and the correction value to substantially obtain the output log value. become.

FIG. 3 gives the output logarithmic value of the input value X to the base m with the first desired precision. 1 is a flowchart showing the configuration of a method of the present invention. The input value A similar value a=mod(x) is obtained (304), where a is a predetermined set of values, that is, a selection is selected from a set of values determined by the rounding algorithm used.

At least the memory device and the approximation a are generally configured as follows: a first approximation (FIV ) (306).

i.e. has a predetermined base p, has a second predetermined precision, and has at least one The first logarithm value of a [log, (aN is generally RO M is used for memory devices, and various types including EFROM are used here. A ROM is selected and the desired precision is preselected. The base p of the first logarithm is a third logarithmic value (In (p)]. And the first logarithm value is (log, (a)) is scaled by multiplying by (In (p)), and the intermediate product FIV = (log , (a))(In(p)) is essentially obtained, which is the natural base logarithm of the approximation a. and has at least a second predetermined precision and a base C, substantially the mouth n (a)] and is represented by at least one electrical signal. In case p=e , the first intermediate value is Ina, thus simplifying the determination of FIV. second Although the predetermined precision is generally selectable, less than the desired precision of the output log value is Common. However, the accuracy of FIV is less than the desired accuracy of the desired output log value. It is equivalent to

At least the input value X and the approximation value a are used to generate the error value C ( 308). Generally, the input value is divided by an approximation value to obtain a first quotient x/a; This first quotient is represented by at least one electrical signal, and 1 is subtracted from this first quotient. , an error value C is substantially obtained, which C is determined by at least one electrical signal. and C is substantially equal to (x/a - 1') or c=(x- An equivalent decision can be made such that a)/a.

At least the error value C is used to calculate the correction value (310), where the error value C is used to calculate the value (SV) of a given number of terms in a given mathematical series, and this A given number of terms is represented by at least one electrical signal and is a given number of terms in a given mathematical series. The value SV with a constant number of terms converts the desired bit-precision value of the output log value into bits of the second log value. greater than or equal to 1 minus the second quotient divided by the precision number, and A given mathematical series of is essentially given by: where C is the error value and be. Also, a given number of terms in this given mathematical series can be estimated and added to reduce A correction value (CV) represented by at least one electrical signal is substantially obtained: e.g. For example, if 24-bit precision is desired for the output logarithm value and the first function generator is When providing 8-bit precision, the SV is generally +(24/8)-11 or 2 is chosen to be equal to .

At least the first intermediate value and the correction value are represented by at least one electrical signal. (312). first intermediate value and The correction values are effectively added to obtain the third sum of the following equation: In x - FIV + CV - Clog, (aN (In (pH + cV) This third sum is the natural value of the input value is the base logarithmic value (in (X)) and is represented by at least one electrical signal. Ru. The natural base logarithm value (in (x)) of the input value X and the desired base of the output logarithm value The 3 quotient is calculated with the natural base logarithm of m [In (m)], and essentially the following formula % formula %: The output logarithmic value (log, , (x)) with the desired base is Represented by a signal. The natural base logarithm value of m is from a preselected ROM by preselecting m of It can be obtained by other methods if necessary. Again, as mentioned above, the natural Note that the use of base logarithms simplifies the step-by-step procedure. .

It will be appreciated that in a preferred embodiment, the invention is implemented entirely in software. That's it. It will also be clear to those skilled in the art that the order of the steps in this method can be varied. Ru. For example, the error value can be calculated before the first intermediate value.

ILI7A abstract Using a function generator that generates the first logarithm of the approximate value (1o 2) of the input value, The logarithm of the value is calculated and scaled to the natural base logarithm (where the first logarithm is the natural base logarithm). ) and is combined with the correction value (110) to obtain the natural base logarithm of the input value.

An error value associated with the input value and the approximation value is generated (104), and the error value is It is used in the power series to determine the correction value (l　O8). Obtained natural base pair The number is scaled by the natural base logarithm of the desired base of the desired output logarithm, thereby Determine the desired output logarithm of the input value with the desired base.

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Claims (10)

[Claims] 1. Process the input value so that the output logarithmic value has the desired base. An apparatus for providing a force log value, the apparatus comprising: A) generating an approximate value in response to said input value; a correction means, wherein the approximation is correction means selected from a predetermined set of values; B) determining a first intermediate value using the approximate value in response to the modifying means; 1 function generating means; C) in response to the input value and the correction means; Error generating means for generating an error value using the approximate value, the error generating means The means includes at least: 1) in response to the input value and the modifying means; and the approximate value to obtain a first quotient between the input value and the approximate value. and 2) in response to said first dividing means, using said first quotient to calculate an error value. , wherein the error value is substantially the difference between the first quotient and the numerical value 1. an error generating means including a subtracting means; D) a correction recommendation for determining a correction value using the error value in response to the error generating means; and E) in response to the first function generating means and the correction estimating means. , a synthesizing means for obtaining an output logarithm value using the first intermediate value and the correction value; The synthesizing means responsive to the first function generating means and the correction estimating means further comprises: In: 1) In response to the first function generating means and the correction estimating means, the correction value is and a second addition means for determining a third equation between and the first intermediate value; 2) a fourth function that determines a third log value in response to the desired base of the output log value; generating means, wherein the third logarithmic value is substantially a natural base of a desired base of the output logarithmic value; a fourth function generating means that is a base logarithm; and 3) In response to the second addition means and the fourth function generation means, the third and previous steps are performed. a second division means for determining a second quotient with the third logarithm value, The second quotient with the two logarithmic values is substantially the second quotient which is the output logarithmic value with the desired base. 2 division means; synthetic means including; A device characterized by comprising: 2. The first function generating means is configured to: A) respond to the modifying means to generate the first function; a second function generating means for determining a first logarithm value having a predetermined base using a similar value; B ) a third function generator for determining a second log value in response to the predetermined base of the first log value; and C) responsive to the second function generating means and the third function generating means; scale the first logarithmic value using the second logarithmic value to realize the first intermediate value. a qualitatively determining first scaling means, wherein the second logarithm value is substantially equal to the first logarithm value; a natural base logarithm of the predetermined base of values, and the first scaling means multiplying the value by the second logarithm value to substantially obtain the first intermediate value; first scaling means including calculation means; 2. The apparatus of claim 1, further comprising: 3. The correction estimating means responsive to the error generating means includes at least: Then, using the error value and the predetermined number of terms of the predetermined mathematical series, calculate the predetermined number of terms. substantially estimate, and substantially add this estimated predetermined number of terms, and substantially determining means for determining a second sum which is the correction value, wherein the predetermined number of terms is the predetermined number of terms; is a subset of terms from the mathematical series, and the number of terms in this subset is , the bit precision value of the output log value is divided by the bit precision value of the first log value. selected to be greater than or equal to the quotient minus 1, and Said predetermined mathematical series is substantially a power series, and this power series consists of 1 and an error value. is essentially a Taylor series for the first degree natural logarithm of es) and is given by the following formula: ▲There are mathematical formulas, chemical formulas, tables, etc.▼ however, c is the error value, The determining means at least estimating means for substantially estimating the predetermined number of terms of a predetermined mathematical series; the predetermined number of terms of the mathematical series are substantially added to substantially the correction value; a first addition means for obtaining a second sum; determining means; 2. The apparatus of claim 1, further comprising: 4. A high-performance device that determines the output logarithm of an input value so that the output logarithm has the desired base. A) a correction method for generating an approximate value in response to the input value; round, the approximation is predetermined by the selected rounding algorithm. corrective means selected from a predetermined set of values that are a set of values that are B) determining a first intermediate value using the approximate value in response to the modifying means; 1 function generating means; C) in response to the input value and the correction means; Error generating means for generating an error value using the approximate value, the error generating means The means includes at least: 1) in response to the input value and the modifying means; and the approximation value to obtain a second quotient of the input value divided by the approximation value. 2 dividing means; and 2) in response to said second dividing means, using said second quotient; subtraction means for determining a difference value, said error value being substantially the sum of said second quotient and the numerical value 1; an error generation means including a subtraction means that is the difference between; D) a correction recommendation for determining a correction value using the error value in response to the error generating means; fixed means; E) in response to the first function generating means and the correction estimating means, the first intermediate value and a first addition means for adding the correction value to obtain a second intermediate value; F) in response to said desired base of said output logarithm, substantially said output logarithm; a second function generating means for determining a second logarithm value that is the natural logarithm of the desired base; and G ) in response to the first addition means and the second function generation means; a first division means for calculating a first quotient with the second logarithm value, the first quotient being substantially , the output logarithmic value having the desired base; A high-performance digital processing device characterized by: 5. The first function generating means is configured to: A) respond to the modifying means to generate the first function; A third function generating means for determining a third logarithmic value having a predetermined base using a similar value; B ) generating a fourth function for determining a fourth logarithmic value in response to the predetermined base of the third logarithmic value; means, wherein the fourth logarithmic value is substantially a natural base of the third logarithmic value. a fourth function generating means that is a base logarithm; and C) In response to the third function generating means and the fourth function generating means, the fourth pair A step of scaling the third logarithmic value using a numerical value to substantially obtain the first intermediate value. 1 scaling means, the first scaling means multiplying the third logarithmic value by the fourth logarithmic value. a first multiplication means for substantially obtaining said first intermediate value; 5. The apparatus according to claim 4, further comprising: 1 scaling means. 6. The correction estimating means responsive to the error generating means includes at least: Then, using the error value and the predetermined number of terms of the predetermined mathematical series, calculate the predetermined number of terms. substantially estimate, and substantially add this estimated predetermined number of terms, and substantially determining means for determining a second sum that is the correction value, the determining means for determining the second sum that is the correction value, The given number of terms is a subset of the terms from the given mathematical series, and The number of terms in the subset is the bit-precision value of the output log value of the first log value. greater than or equal to 1 minus the quotient divided by a bit-precision number and said predetermined mathematical series is substantially a power series; The power series is essentially a Taylor series for the first natural logarithm of 1 and the error value. Yes, given by the following formula: ▲There are mathematical formulas, chemical formulas, tables, etc.▼ However, c is a determining means that is an error value; 5. The device according to claim 4, further comprising: 7. The determining means at least substantially determines the predetermined number of terms of the predetermined mathematical series. estimating means for estimating the predetermined number of terms of the predetermined mathematical series; and a first addition means for obtaining a second sum which is substantially the correction value. 5. The device of claim 4, characterized in that: 8. Process an input value x to obtain an output logarithm value {log m(x)}, whose output logarithmic value is a device with the desired precision: A) Modifying means for generating an approximate value a in response to the input value x, the approximate value a is selected from a given set of values that is at least a given set of rounding algorithm values. corrective measures to be taken; B) in response to said modifying means, utilizing said approximation to substantially {ln(a)} a first function generating means for determining a first intermediate value; C) In response to the input value x and the correction means, the input value x and the approximate value a An error generating means for generating an error value c using at least: 1) in response to said input value x and said modifying means; The first quotient x/a obtained by dividing the input value x by the approximate value a is calculated using the approximate value a. and 2) in response to said first dividing means, said first dividing means for calculating said first quotient x/ Using a and the numerical value 1, the subtraction that essentially calculates the error value c = 1 {(x/a)-1} means of calculation; error generating means including; D) Correction of generating a correction value using the error value c in response to the error generating means. The estimating means, the corrected estimating means responsive to the error generating means at least: In response to the error generating means, the error value and a predetermined number of terms of a predetermined mathematical series are calculated. to estimate a predetermined number of terms, and to substantially add this estimated predetermined number of terms. determining means for calculating a second sum which is substantially the correction value, the predetermined number of terms of the mathematical series is a subset of terms from the predetermined mathematical series; , and the number of terms in this subset is the bit-precision value of the output log value. Is greater than or equal to the value obtained by dividing the first logarithm value by the bit-precision value and subtracting it from 1. or equal, and said predetermined mathematical series is substantially equal to Given: ▲Mathematical formulas, chemical formulas, tables, etc.▼ However, c is the error value determining means Correction estimation means including; E) in response to the first function generating means and the correction estimating means, the first intermediate value and the correction value to obtain the output logarithm value {1ogm (x)}; A device characterized by comprising: 9. The first function generating means is configured to: A) respond to the modifying means to generate the first function; Using the approximation a, the a second function generating means for determining a logarithmic value {logp(a)}, the predetermined positive The accuracy is less than or equal to the desired accuracy of the output log value, and a second function generator whose precision is at least equal to said desired precision of said output log value; stage; B) in response to the desired base of the first logarithmic value; third function generating means for determining a third natural base logarithm value ln(p)}; and C) In response to the second function generating means and the third function generating means, the first pair A first intermediate value having at least said predetermined precision and base e by scaling a numerical value, all That is, it is essentially a first scaling means for determining {ln(a)}, and the first scaling means The step is at least a base of the first logarithm for the first logarithm value {logp(3)}. Multiplying by a coefficient substantially equal to {ln(p)}, which is the natural base logarithm of p, a second multiplication means for substantially determining the first intermediate value, the determining means at least estimating means for substantially estimating the predetermined number of terms of the mathematical series; and the mathematical series. substantially adding the predetermined number of terms to obtain a second sum that is substantially the correction value. a first scaling means including a first addition means; 9. The apparatus of claim 8, further comprising: 10. Said synthesis means comprises at least: A) In response to the first function generating means and the correction estimating means, the first intermediate value and a second addition means for calculating a second sum using the correction value, the second sum being a second addition means which is substantially the natural base logarithmic value {ln(x)} of said input value x; call B) in response to the second addition means and the third function generation means, the the natural base logarithm {ln(x)}; and the natural logarithm of the desired base m of the output logarithm {l Determine the third quotient with n(m)} and use the following formula: {ln(x)}/{ln(m)}={logm(x)}, that is, the desired base further comprising: fourth division means for obtaining the output logarithm value {logm(x)} having 9. The device according to claim 8, comprising: