This technology relates to an information processing apparatus, a program, and an information processing method for performing numerical arithmetic operations. This application claims priority on the basis of Japanese Patent Application No. 2019-214678, filed on Nov. 27, 2019, in Japan, which is incorporated by reference herein.
Numerical arithmetic operation is a method for numerically solving a problem which is difficult to solve analytically, and a computer floating-point arithmetic is usually used for the arithmetic operation. The IEEE (Institute of Electrical and Electronic Engineers) 754 is a standard for floating-point arithmetic. At present, most numerical arithmetic operation programs are implemented in accordance with the IEEE 754 standard, and their assets are enormous.
Floating-point arithmetic has several drawbacks, including overflow/underflow, digit loss, information loss, leftover, and round-off error. For this reason, although there has been no major problem even in double precision floating-point arithmetic at the level of there is a concern that the effective digit may be insufficient at the level of nm. Furthermore, it is assumed that arithmetic operation errors are likely to occur if a series of higher order polynomials is handled.
At present, arithmetics considering errors (validated numerics) are known, and there is an arithmetic using a software to perform arithmetic operations while confirming the range of the true solution so as to yield a result guaranteed to be mathematically true for a problem of continuity in mathematics. The validation by this software is intended to deal with the problem of continuous numbers by inserting a true value to be operated into a section both ends of which are floating-point numbers which can be handled by a processor. The four arithmetic operations for handling these intervals in the same way as numbers are called interval arithmetic, and those that perform the interval arithmetic using floating-point numbers are called machine interval arithmetic. However, the interval arithmetic requires a large amount of arithmetic operations and is complicated because it denotes real numbers as ranges and exactly performs the four arithmetic operations on the ranges.
[Patent Reference 1] Japanese Unexamined Patent Application Publication No. 2004-05395
The present technology has been proposed in view of such conventional circumstances, and provides an information processing apparatus, a program, and an information processing method that can easily perform validated numerics.
As a result of intensive examination, the present inventors have found that the above problem can be solved by introducing a (high precision) computer number that uses a definite number and an uncertain number separately to expresses a number belonging to a subset of real numbers which can be expressed in n-ary numbers with a radix n.
Specifically, an information processing apparatus according to the present technology includes: an input unit for inputting a high-precision computer number in which real numbers A and B are respectively defined by the following formulas (11) and (12); a storage unit for storing the high-precision computer number; and an arithmetic unit that performs an arithmetic operation by using definite numbers a1 and b1 of the high-precision computer numbers stored in the storage unit, determines an extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13), and storing, in the storage unit, high-precision computer numbers as the result of the arithmetic operation.
Moreover, a program according to the present technology causes a computer to execute a processing including: an input step of inputting a high-precision computer number in which real numbers A and B are respectively defined by the following formulas (11) and (12); a storing step of storing the high-precision computer number in a storage unit; and an arithmetic step of performing an arithmetic operation by using definite numbers a1 and b1 of the high-precision computer number stored in the storage unit, determining an extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13), and storing, in the storage unit, high-precision computer numbers as a result of the arithmetic operation.
Furthermore, an information processing method according to the present technology includes: an input step of inputting a high-precision computer number in which real numbers A and B are respectively defined by the following formulas (11) and (12); a storing step of storing the high-precision computer number in a storage unit; and an arithmetic step of performing an arithmetic operation by using definite numbers a1 and b1 of the high-precision computer number stored in the storage unit, determining an extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13), and storing, in the storage unit, high-precision computer numbers as a result of the arithmetic operation.
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
(A,B)=(a,b)1+(a,b)2,−Cγ+γ′≤(a,b)2<Cγ+γ′ (13)
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha+ha′ and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha+ha′ and |b1|≤Chb+hb′; ha and hb respectively denote high order maxes that are integers; ha′ and hb′ respectively denote decimal digits that are decimals of 0≤ha′<1 and 0≤hb′<1; ea+ea′, eb eb′, and γ+γ′ denote extended absolute effective digits that are extended digits; ea, eb, and γ denote absolute effective digits that are integers; ea, eb′, and r′ respectively denote decimal digits that are decimals of 0≤ea′<1, 0≤eb′<1, and 0≤r′<1; (A, B) is the arithmetic operation result of the real numbers A and B; (a, b)1 is the arithmetic operation result of the definite numbers a1 and b1; and (a, b)2 is the arithmetic operation result of the uncertain numbers a2 and b2.
According to this technology, it is possible to easily perform validated numerics. While interval arithmetic treats numbers as a range, the computer number is just a kind of number and the uncertain number is not accurately determined and can take any values within the range of absolute effective digit, so that the idea of computer number is different from that of interval arithmetic.
Hereinafter, a first embodiment of the present technology will be described in detail in the following order with reference to the drawings. It should be noted that the present technology is not limited to the following embodiment, and various modifications may be made without departing from the scope of the present technology.
1-1. Information processing apparatus
1-2. Arithmetic operation of computer numbers
1-3. Examples
Functional Configuration
A=a1+a2,|a1|≤Ch
A=b1+b2,|b1|≤Ch
(A,B)=(a,b)1+(a,b)2,−Cγ≤(a,b)2<Cγ (3)
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha and hb respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha and |b1|≤Chb, and ha and hb respectively denote high order maxes that are integers; ea, eb and γ denote absolute effective digits that are integers; and (A, B) is an arithmetic operation result of the real numbers A and B, (a, b)1 is an arithmetic operation result of the definite numbers a1 and b1, and (a, b)2 is an arithmetic operation result of the uncertain numbers a2 and b2.
The radix C of the computer number is not particularly limited, but is preferably 2, 8, or 16. Thus, it is possible to verify the arithmetic operation of binary, octal or hexadecimal numbers used in general-purpose computers.
The input unit 11 inputs computer numbers in which real numbers A and B are respectively defined by the following formulas (1) and (2).
A=a1+a2,|a1|≤Ch
A=b1+b2,|b1|≤Ch
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha and hb respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha and |b1|≤Chb, and ha and hb respectively denote high order maxes that are integers; and ea, eb and γ denote absolute effective digits that are integers.
Specifically, the input unit 11 inputs computer numbers for which a sign, high order max, low order max, absolute effective digit, and number of arrays are set.
The storage unit 12 stores computer numbers and is a so-called buffer. The buffer is reserved based on the high order max and the low order max inputted by the input unit 11.
The arithmetic unit 13 performs an arithmetic operation by using the definite numbers a1 and b1 of computer numbers stored in the storage unit 12, determines an absolute effective digit r so as to satisfy the following formula (3), and stores, in the storage unit 12, computer numbers as the result of the arithmetic operation.
(A,B)=(a,b)1+(a,b)2,−Cγ≤(a,b)2<Cγ (3)
Herein C denotes a radix; γ is an exponent and is an absolute effective digit; and (A, B) is an arithmetic operation result of the real numbers A and B, (a, b)1 is an arithmetic operation result of the definite numbers a1 and b1, and (a, b)2 is an arithmetic operation result of the uncertain numbers a2 and b2.
The arithmetic operation of computer numbers is performed on the assumption that the real numbers A and B fall within the range of a1−Cea≤A<a1+Cea and b1−Ceb≤B<b1+Ceb, respectively, so that the arithmetic operation result is also a computer number and thus satisfies the formula (3). The arithmetic operation of the computer number is divided into the definite numbers a1, b1 and the uncertain numbers a2, b2, the definite numbers a1, b1 are calculated as they are, and the absolute effective digit r of the arithmetic operation result is determined according to various conditions.
In the arithmetic operation of computer numbers, the maximum value of the absolute effective digit can be specified. In particular, for division and functions such as root, sin, and cos, the absolute effective digit can be specified. Thus, the numerical value of the absolute effective digit or more can be guaranteed.
The arithmetic operation of computer numbers is also considered to be a special pattern of interval arithmetic. However, the difference is that the interval arithmetic represents a real number as a range and performs four arithmetic operations on the range exactly, while the four arithmetic operations of computer numbers controls errors by the absolute effective digit. In addition, since the arithmetic operation of computer numbers is considered to be applied after determining a system of numbers instead of real numbers, the starting point thereof is different from the concept of interval arithmetic. Therefore, the arithmetic operation itself of computer numbers is simpler than interval arithmetic, and the arithmetic operation is simplified by using the absolute effective digit as an evaluation of errors (unclear parts).
According to such an information processing apparatus, only the part requiring the absolute effective digit can be calculated by a computer number arithmetic operation library, and the absolute effective digit of the arithmetic operation result can be confirmed, so that validated numerics can be easily performed.
The information processing apparatus preferably further includes a control unit 14 for converting a numerical arithmetic operation program for real numbers into a numerical arithmetic operation program for computer numbers, executing the numerical arithmetic operation program for computer numbers, and storing, in the storage unit 12, computer numbers obtained by the arithmetic operation in the arithmetic unit 13. Thus, the history of the absolute effective digit of each variable can be maintained during the execution of the numerical arithmetic operation program for computer numbers, and it is possible to check whether the required absolute effective digit can be finally obtained. It is also possible to inversely calculate the absolute effective digit of the value of each variable required for an arithmetic operation of a desired absolute effective digit.
Hardware Configuration
The CPU 21 is capable of executing processes of the arithmetic unit 13 and the control unit 14 in the functional configuration shown in
The GPU 22 has a video memory (VRAM), can perform drawing processing and arithmetic operation processing in response to a request from the CPU 21, and may calculate computer numbers by the computer number arithmetic operation library. The ROM 23 is, for example, a read-only nonvolatile memory, and stores information such as constants necessary for the operation of each block of the information processing apparatus 2. The RAM 24 is a volatile memory, and is used not only as a deployment area for an arithmetic program but also as a storage area for temporarily storing intermediate data or the like outputted during the operation of each block of the information processing apparatus 2.
The operation input unit 25 can control the function of the input unit 11 in the functional configuration shown in
The storage 26 records a numerical arithmetic operation program or the like deployed in the RAM 24. The storage 26 may be an HDD (Hard disk drive), an SSD (Solid State Drive), or an optical drive, among others. The input/output interface 27 can output images generated by the GPU 22 to a display device.
In such a hardware configuration, the functional configuration shown in
Hereinafter, the arithmetic operation method will be described with the radix of the computer number being 2 (C=2).
Addition
Assume that a real number A and a real number B are computer numbers defined by the formulas (1) and (2), respectively.
A=a1+a2,|a1|≤Ch
A=b1+b2,|b1|≤Ch
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha and hb respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha and |b1|≤Chb, and ha and hb respectively denote high order maxes that are integers; and ea, eb and γ denote absolute effective digits that are integers.
Hereinafter, m, n, and r will be used for explanation instead of the absolute effective digits ea, eb, and γ, respectively.
A+B=a1+a2+b1+b2=(a1+b1)+(a2βb2)⇒(a1+b1)−Cm−Cn≤A+B<(a1+b1)+Cm+Cn
Assume that r is the smallest integer that satisfies Cm+Cn≤Cr.
If m=n, then r=m+1; if m≠n, then r=max (m, n)+1, so r=max (m, n)+1.
A+B=(a1+b1)++B)2
(a1+b1) is a definite number, (A+B)2 is an uncertain number, −Cr≤(A+B)2<Cr is satisfied, and r is the absolute effective digit. The minimum order of the definite number (a1+b1) may be less than the absolute effective digit. A description will be given by classifying patterns according to the sign of (a1+b1).
(a1+b1)−Cr≤A+B<(a1+b1)+Cr⇒(α+β)+γ−Cr≤A+B<(α+β)+γ+Cr
If γ>0 ((a1+b1)>0),
then (α+β)−Cr−1≤A+B<(α+β)+Cr−1.
If γ<0 ((a1+b1)<0),
then (α+β)−Cr+1≤A+B<(α+β)+Cr+1.
Thus, (α+β)−Cr+1≤A+B<(α+β)+Cr+1 holds, (α+β) is a definite number, and the absolute effective digit is r+1.
If γ=0, then (α+β)−Cr≤A+B<(α+β)+Cr holds, (α+β) is a definite number, and the absolute effective digit is r.
Multiplication
A*B=(a1+a2)*(b1+b2)=a1b1+a2b1+a1b2a2b2
Assume that A and B are both positive.
a1≤Cα; α is the smallest integer that satisfies the inequality. It is preferably log2|a1|.
b1≤Cβ; β is the smallest integer that satisfies the inequality. It is preferably log2|b1|.
−CmCβ≤a2b1<CmCβ,
−CaCm≤a1b2<CaCm, and
−CmCn<a2b2≤CmCn,
therefore,
a1b1−CmCβ−Cm−CmCn<A*B<a1b1+CmCβ+CaCm+CmCn⇒a1b1−Cm+β−Cα+m−Cm+n<A*B<+Cm+β+Cα+m+Cm+n
Assume that r is the smallest integer that satisfies Cm+β+Cα+m+Cm+m≤Cr.
A*B=a1b1+(A*B)2
a1b1 is a definite number, (A*B)2 is an uncertain number, −Cr≤(A*B)2<Cr is satisfied, and r is the absolute effective digit. The minimum order of the definite number a1b1 may be less than the effective digit.
The definite number and the absolute effective digit considering the meaningless digit part are the same as the positive case in addition.
Assuming that r is the smallest integer that satisfies Cm+β+Cα+m+Cm+n≤Cr, then Cm+β+Cα+m+Cm+n=Cr holds only in pattern 3. Further, r that satisfies Cm+β+Cα+m+Cm+n≤Cr is the maximum value+2 (r=max (m, n)+2) if (m+β), (α+n), and (m+n) are pattern 1 or pattern 2, or is the maximum value+1 (r=max (m, n)+1) otherwise. Moreover, r that satisfies Cm+β+Cα+m+Cm+n<Cr is the maximum value+2 (r=max (m, n)+2) when (m+β), (α+n) and (m+n) are pattern 1, pattern 2 or pattern 3, or is the maximum value+1 (r=max (m, n)+1) otherwise.
Next, arithmetic operation examples of software using the computer numbers described above will be described.
As shown in
In the multiplication, addition, and division of computer numbers shown in
Arithmetic operation of Rump formula
The value of f (a, b) obtained by substituting a=77617 and b=33096 into the following formula devised by S. M. Rump is evaluated.
f(x,y)=333.75y6+x2(11x2y2−y6−121y4−2)+55y8+x/2y
When this arithmetic operation is executed on the IBM mainframe S/370 with different operational precision, the following results are obtained.
Single precision (approximately 8 decimal digits): f (x, y)≈1.1720603 . . . .
Double precision (approximately 17 decimal digits): f (x, y)≈1.17206039400531 . . . .
Extended precision (approximately 34 decimal digits): f (x, y)≈1.1720603940053178 . . . .
The true value is f (a, b)=−0.827386 . . . ; therefore the arithmetic operations on S/370 have wrong results whose signs are not even correct.
f1=333.75y6+x2*11x2y2+55y8+x/2y
f2=x2*y6+x2*121y4+x2*2
f=f1−f2
Arithmetic Operation of Series
1/(1−x)=1−x+x2+ . . . +(−1)nxn . . . (−1<x<1)
A formula of Maclaurin expansion is evaluated. If terms −x+x2+ . . . +(−1)nxn . . . are added sequentially to 1−1/(1−x), then the result should approach 0.
Table 1 shows the results of arithmetic operations using double precision floating-point. There were no differences in errors depending on S. It is revealed that, since each term of the intermediate series is a sum of values smaller than the initial value when calculating the sum of the series, the effective digit is smaller than the initial value. It might be difficult to calculate the sum of series even when double precision floating-point numbers are used. On the other hand, in the arithmetic operation of computer numbers, the result was 0 when calculated with the effective digit being −500.
Hereinafter, a second embodiment of the present technology will be described in detail in the following order with reference to the drawings. It should be noted that the present technology is not limited to the following embodiment, and various modifications may be made without departing from the scope of the present technology.
2-1. Information Processing Apparatus
2-2. Arithmetic Operation of High-Precision Computer Numbers
2-3. Example
In the arithmetic operation of the multiplication of computer numbers according to the first embodiment, since the absolute effective digit increases by about three digits, the accuracy of computer numbers decreases rapidly when multiplications are repeated. Further, in additions with different absolute effective digit, since the absolute effective digit always increases by one, the accuracy surely decreases when additions are repeated.
Therefore, in the second embodiment, high-precision computer numbers are used instead of computer numbers in the first embodiment. The high-precision computer numbers uses extended digits (extended high order max and extended absolute effective digit) (α+α′) instead of the digits in the computer number (high order max and absolute effective digit). Herein, a is an integer, and this value is the number of digits. α′ is a decimal digit which is a decimal of 0≤α′<1, and for example, a′ may be a/2n(0≤a<2n).
When generating a high-precision computer number at first, the extended absolute effective digit is an integer. The decimal digit of the extended absolute effective digit can suppress the deterioration of precision in the case where the four arithmetic operations are repeated, and is particularly effective when there is a difference in the absolute effective digits.
Functional Configuration
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
(A,B)=(a,b)1+(a,b)2,−Cγ+γ′≤(a,b)2<Cγ+γ′ (13)
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha+ha′ and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha+ha′ and |b1|≤Chb+hb′; ha and hb respectively denote high order maxes that are integers; ha′ and hb′ respectively denote decimal digits that are decimals of 0≤ha′<1 and 0≤hb′<1; ea+ea, eb eb′, and γ+γ′ denote extended absolute effective digits that are extended digits; ea, eb, and γ denote absolute effective digits that are integers; ea′, eb′, and r′ respectively denote decimal digits that are decimals of 0≤ea′<1, 0≤eb′<1, and 0≤r′<1; (A, B) is the arithmetic operation result of the real numbers A and B; (a, b)1 is the arithmetic operation result of the definite numbers a1 and b1; and (a, b)2 is the arithmetic operation result of the uncertain numbers a2 and b2.
The radix C of the computer number is not particularly limited, but is preferably 2, 8, or 16. Thus, it is possible to verify the arithmetic operation of binary, octal or hexadecimal numbers used in general-purpose computers.
The decimal digits ha′, hb′, ea, eb′, and r′ are preferably ha′/2n (0≤ha′<2n), hb′ 2n (0≤hb′<2m), ea′/2n (0≤ea′<2n), eb′/2n (0≤eb′<2n), r′/2n (0≤r′<2n), respectively, and n is a natural number. n is preferably 32 or less, more preferably 16 or less, and still more preferably 8 or less. Increasing n improves precision, but increases arithmetic operation cost. It should be noted that, when the decimal digits ha′, hb′, ea′, eb′, and r′ are 0, this is the computer number according to the first embodiment.
The input unit 31 inputs the high-precision computer number in which real numbers A and B are respectively defined by the following formulas (11) and (12).
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C denotes a radix; ha+ha′ and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤Cha+ha′ and |b1|≤Chb+hb′, ha and hb respectively denote high order maxes that are integers; ha′ and hb′ respectively denote decimal digits that are decimals of 0≤ha′<1 and 0≤hb′<1; ea+ea′, eb+eb′, and γ+γ′ denote extended absolute effective digits that are extended digits; ea and eb denote absolute effective digits that are integers; ea′ and eb′ respectively denote decimal digits that are decimals of 0≤ea′<1, and 0≤eb′<1;
Specifically, the input unit 31 inputs high-precision computer numbers for which a sign, high order max h that is the most significant digit, decimal digit h′ of the high order max (high order max sub), low order max l that is the least significant digit, the absolute effective digit e (effective digit), the decimal digit e′ of the absolute effective digit (effective digit sub), and number of arrays are set.
The storage unit 32 stores the high-precision computer number and is a so-called buffer. The buffer is reserved based on the high order max h and the low order max l inputted by the input unit 31. For example, as shown in
The arithmetic unit 33 performs an arithmetic operation by using the definite numbers a1 and b1 of the computer numbers stored in the storage unit 32, determines the extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13), and stores, in the storage unit 32, computer numbers as the result of the arithmetic operation.
(A,B)=(a,b)1+(a,b)2,−Cγ+γ′≤(a,b)2<Cγ+γ′ (13)
Herein C denotes a radix; γ+γ′ denotes the extended absolute effective digit, γ denotes the absolute effective digit which is an integer, and γ′ denotes the decimal digit which is a decimal of 0≤γ′<1; (A, B) is an arithmetic operation result of the real numbers A and B, (a, b)1 is an arithmetic operation result of the definite numbers a1 and b1, and (a, b)2 is an arithmetic operation result of the uncertain numbers a2 and b2.
The arithmetic operation of the high-precision computer number is performed by assuming that the real numbers A and B fall within the range of a1−Cea+ea′≤A<a1+Cea+ea′ and b1−Ceb+eb′≤B<b1+Ceb+eb′ respectively, so that the arithmetic operation result is also the high-precision computer number and thus satisfies the formula (13). The arithmetic operation of the high-precision computer number are divided into the definite numbers a1, b1 and the uncertain numbers a2, b2, the definite numbers a1, b1 are calculated as they are, and the extended absolute effective digit (γ+γ′) of the arithmetic operation result is determined according to various conditions. It should be noted that the effective digit k is the smallest integer that satisfies logC|definite number|−extended absolute effective digit (γ+γ′)≤k.
In the arithmetic operation of addition (A+B), the arithmetic unit 33 can set the minimum extended digit satisfying the following formula (14) as the extended absolute effective digit.
Ce
In the arithmetic operation of multiplication (A*B), the arithmetic unit 33 can set the minimum extended digit satisfying the following formula (15) as the extended absolute effective digit.
Ch
In the arithmetic operation of division (B/A), the arithmetic unit evaluates the following formula (18) as the high-precision computer number defined by the following formulas (16) and (17) by using ha+ha′−1/256 and hb+hb′−1/256 instead of ha+ha′ and hb+hb′, respectively, to determine the extended absolute effective digit.
In four arithmetic operations of the high-precision computer number, it is necessary to perform arithmetic operations such as |a|≤infCα+α′ (C: radix, a: definite number), supCα+α′+supCβ+β′≤infCγ+γ′|(C: radix), and infCα+α′−supCβ+β′≥supγ+γ′ (C: radix). Herein, supA denotes the upper limit value of A, and infA denotes the lower limit value of A.
Preferably, the arithmetic unit 33 performs arithmetic operations for |a|≤infCα+α′ by using a list table of values a for α′ when α=0, for supCα+α′+supCβ+β′≤infCγ+γ′ by using a list table of values γ and γ′ for α′, β and β′ when α=0, and for infCα+α′−supCβ+β′≥supCγ+γ′ by using a list table of values γ and γ′ for α′, β and β′ when α=0. Thus, the arithmetic operations of the high-precision computer numbers can be accelerated.
Further, it is preferable that the arithmetic unit 33 trims the definite numbers of supCα+α′+supCβ+β′≤infCγ+γ′ based on the difference of infCγ+γ′−(supCα+α′+supCβ+β′). This can suppress the arithmetic operation cost when the arithmetic operations are repeated. Similarly, for infCα+α′−supCβ+β′≥supCγ+γ′, trimming the definite numbers based on the difference of infCα+α′−supCβ+β′−supCγ+γ′ can suppress the arithmetic operation cost when the arithmetic operations are repeated.
The information processing apparatus preferably further includes a control unit 34 for converting a numerical arithmetic operation program for real numbers into a numerical arithmetic operation program for high-precision computer numbers, executing the numerical arithmetic operation program for high-precision computer numbers, and storing, in the storage unit 32, computer numbers obtained by the arithmetic operation in the arithmetic unit 33. Specifically, it is preferable to convert the floating-point representation to high-precision computer numbers. Thus, the history of the extended absolute effective digit of each variable can be maintained during the execution of the numerical arithmetic operation program for high-precision computer numbers, and it is possible to check whether the required extended absolute effective digit can be finally obtained. It is also possible to inversely calculate the absolute effective digit of the value of each variable required for an arithmetic operation of a desired extended absolute effective digit arithmetic operation.
Such an information processing apparatus can calculate only the part requiring the extended absolute effective digit by the high-precision computer number arithmetic operation library, and confirm the extended absolute effective digit of the arithmetic operation result, so that validated numerics can be easily performed.
Hardware Configuration
The information processing apparatus may have a hardware configuration shown in
Hereinafter, the arithmetic operation method will be described with the radix of the high-precision computer number being 2 (C=2), and the decimal digits ha′, hb′, ea′, eb′, and r′ of the exponents being ha′/28 (0≤ha′<28), hb′/28 (0≤hb ′<28), ea′/28 (0≤ea′<28), eb′/28 (0≤eb′<28), and r′/28 (0≤r′<28), respectively. However, in order to simplify the description, the decimal digits of the exponents are also denoted as ha′, hb′, ea′, eb′, and r′.
If n=8, then, for example, in the relation of supCα+α′+supCβ+β′≤infCγ+γ′, the difference of infCγ+γ′−(supCα+α′+supCβ+β′) does not become 0 except in a specific case, and a gap occurs. Since the arithmetic operation is not affected even if the part below this gap is truncated, trimming processing can be performed, thereby suppressing the arithmetic operation cost when the arithmetic operations are repeated.
The significand stores arrays a of 32 bits, and the maximum number of the arrays a is 127. Since the array a is an unsigned integer value, the maximum size of the significand is 127*32=4064 bits. This can represent 1223 digits in the decimal notation. For example, since a light year is 9.5 trillion km<10 trillion km=1013 km=1019 mm=1025 nm, assuming the size of the universe is 137 light years, or about 100 light years, 100 light years=1010 light years=1035 nm, so that 35 digits (decimal number) are sufficient to express the size of the universe in nm. In relation to this matter, by using the high-precision computer numbers, arithmetic operation of very large order can be performed.
Signed 16 bit integers can be used for the high order max (most significant digit) h, the low order max (least significant digit) 1, and the absolute effective digit (effective digit) e. A signed 16 bit integer is ±32768, so a sufficient representation is possible. It should be noted that, when the absolute effective digit e is −32768 (1000000000000000 in the binary notation), the error is regarded as 0.
For the decimal digit h′ of the high order max and the decimal digit e′ of the absolute effective digit, an unsigned 8-bit integer such as an unsigned char variable in the C language can be used. When the decimal digit h′ of the high order max is 0, there are two meanings. If h=1, then h′ is 0 as it is, but if h>1, then h′=256 and the high order max is (h+1).
Treatment of the Uncertain Number 0
The range of an uncertain number is expressed as −Ce+e′≤(uncertain number)<Ce+e′ by using the extended absolute effective digit. If the absolute effective digit e is −∞, then the possible width of the uncertain number is infinitely close to 0. In the representation of the high-precision computer numbers in a computer, the minimum value of the absolute effective digit e is −32768, and this value is treated as −∞. If e→−∞, then the uncertain number is defined as 0. The value of the absolute effective digit e that can be taken in the arithmetic operation of the high-precision computer number is a value in the range from −32767 to 32767, and when it exceeds this range, it is treated as an arithmetic operation error.
0 in High-Precision Computer Number
The representation of 0 in the high-precision computer number is defined as follows.
s (sign)→0
h (high order max)−0
hd (decimal digit of the high order max)→0
l (low order max)→0
e (absolute effective digit)→calculated value
ed (decimal digit of the absolute effective digit)→calculated value
array a→a [0]=0
The size of the array a in high-precision computer numbers does not change, but the minimum value of the array a is set to 0 as an exception. In computer numbers other than 0, the 0th bit of the array a is always 1.
Extended Digit Table
Decimal Digit h′ of Extended High Order Max
As shown in
The first 9 bits of the extracted 32 bits are used to calculate a′ whose lower limit value is larger than the value obtained by dividing the first 9 bits by 256. This process is performed by referring to an 8 bit-α′ table which outputs a′ corresponding to an 8-bit integer.
Addition
Assume that a real number A and a real number B are the high-precision computer numbers defined by the formulas (11) and (12), respectively.
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
wherein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C=2; ha+ha′ and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤2ha+ha′ and |b1|≤2hb+hb′; ha and hb respectively denote high order maxes that are integers, ha′ and hb′ respectively denote decimal digits that are ha′/256 and hb′/256; ea+ea′ and eb+eb′ denote extended absolute effective digits; ea and eb denote absolute effective digits that are integers; and ea′ and eb′ respectively denote decimal digits which are decimals of ea′/256 and eb′/256.
A+B is expressed by the following formulas (21) to (23).
A+B=a1+a2+b1+b2=a1+b1+a2+b2 (21)
(a1+b1)−Ce
Ce
The minimum extended digit that satisfies the formula (23) is determined. The values such as Cea+ea′, Ceb+eb′ and Cγ+γ′ are expressed by high-precision computer numbers, and supCea+ea′+supCeb+eb′≤infCγ+γ′ is determined. Herein, supA denotes the upper limit value of A, and infA denotes the lower limit value of A.
The case when the uncertain number is 0
If the uncertain number b2 of the real number B is 0 in A+B shown in the formula (21), then the inequality shown in the formula (22) becomes the following formula (24).
(a1+b1)−Ce
Further, if the uncertain number a2 of the real number A is 0, the uncertain number of A+B becomes 0. In this case, the trimming of the lower digits is not performed.
Extended Digit Addition Table
The arithmetic operation of supCea+ea′+supCeb+eb′≤infCγ+γ′ as shown in the formula (23) uses an extended digit addition table which outputs the minimum extended digit (γ+γ′) satisfying Cα+Cβ+β′≤Cγ+γ for Cα and Cβ+β′. Herein, 0≥β, and α′ and β′ are α′/28 and β′/28, respectively.
It is understood that in the definite number, the position of γ+β is the position of the original digits, and the lower digits that have been shifted from this position by the digits of the difference may be truncated. That is, the difference is absorbed into the offset.
In Cα+α′+Cβ+β′≤Cγ+γ′, if (α+α′) coincides with (β+β′), then γ=2α, γ′=α′, and the offset value becomes ∞. This is actually expressed as 65535 (bit16max in the program), which is the maximum value of 16 bit integers. In this case, since the difference becomes 0, the digits of the calculated value becomes large. In the processing in this case, the same processing as the division described later is performed, so that (γ+γ′)+1/256 is set to the extended absolute effective digit and the digits of the calculated value is restricted by using the difference between Cγ+γ′ and Cγ+γ+1/256.
The extended digit addition table is classified by 0 (integers between 0 and −32) and outputs γ, γ′ and the digits of the difference for α′ and β′.
(1) If β=0, α′≥β′, then the extended digit (γ+γ′) satisfying supCα′+supCβ+β′≤infCγ+γ′ is determined, and the difference (infCγ+γ′−(supCα′+supCβ+β′)) is evaluated to create a table of α′, β′, γ, γ′, and the digits of the difference.
(2) If β is −1 to −9, then the table of α′, β′, γ, γ′, and the digits of the difference is created by determining the extended digit (γ+γ′) satisfying supCα′+supCβ+β′≤infCγ+γ′ and evaluating the difference (infCγ+γ′−(supCα′+supCβ+β′)) as in (1).
(3) If β is −10 to −32, then γ=−0 and γ′=α′+1, regardless of the value of β′. However, if α′=255, then γ=+1 and γ′=0. By evaluating the difference (infCγ+γ′−(supCα′+supCβ+β′)), the table of α′, β′, γ, γ′, and the digits of the difference is created.
(4) If β is −33 or less, then γ=−β and γ′=α′+1, regardless of the value of β′ as in (3). However, if α′=255, then γ=+1 and γ′=0. By evaluating the difference (infCγ+γ′−(supCα′+supCβ+β′)), the table of α′, β′, γ, γ′, and the digits of the difference is created. The digits of the difference is 9 or less.
(5) If β=0 and α′=β′, then the digits of the difference is 65535. This is the maximum value of unsigned single-precision integers. In this case, trimming of the array a is not performed.
22/256+20=2.0054299011 . . .
21+2/256=2.010859802 . . .
Further, since 21+1/256=2.005422550 . . . , 21+2/256 is the minimum value.
Further, the difference (21+2/256−(22/256+20)) is 0.0054299009 . . . . Multiplying this value by 256 yields 1.3900546304. This indicates that when the value of the difference is shifted to the left by 8 bits, the head position becomes the position of γ. Therefore, it is shifted to the right by 8 bits from γ so as to set the digits of the difference less than 9 to be 0.
Multiplication
Assume that a real number A and a real number B are the high-precision computer numbers defined by the formulas (11) and (12), respectively.
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C=2; ha+ha′ and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤2ha+ha′ and |b1|≤2hb+hb′; ha and hb respectively denote high order maxes that are integers, ha′ and hb′ respectively denote decimal digits that are ha′/256 and hb′/256; ea+ea′ and eb+eb′ denote extended absolute effective digits; ea and eb denote absolute effective digits that are integers; and ea′ and eb′ respectively denote decimal digits which are decimals of ea′/256 and eb′/256.
A*B is expressed by the following formula (31).
A×B=(a1+a2)×(b1+b2)=a1b1+a2b1+a1b2+a2b2 (31)
Hereinafter, ha, ha′, hb and hb′ are denoted by ma, ma′, mb and mb′, respectively.
(1) If A>0 and B>0, then a1≤Cma+ma′, b1≤Cmb+mb′, and the following formulas (32) to (34) are satisfied.
−Ce
−Cm
−Ce
The formulas (32) to (34) can be rearranged to be expressed as the following formulas (35) to (37).
−Cm
−Cm
−Ce
According to the following formula (38), the formula (31) is expressed as the following formula (39), and the extended digit satisfying the formula (40) is just to be determined.
C′=Cm
a1b1−C′<A×B<a1b1+C′ (39)
C′≤Cr+r′ (40)
In order to determine the extended digit, the minimum extended digit satisfying the following formula (41) is just to be determined.
Cα+α′+Cβ+β′≤Cγ+γ′ (41)
If the uncertain number b2 of the real number B is 0, then the formula (38) becomes the following formula (42).
Cγ+γ′=Cm
If both the uncertain numbers of the real numbers A and B are 0, then γ→−∞, and the value of the absolute effective digit e is −32768 (1000000000000000). If the uncertain number of either of the real numbers A and B is 0, then the trimming processing of the lower digit is not performed.
(2) If A>0 and B<0, then a1≤Cma+ma′, −b1≤Cmb+mb′, and the following formulas (43) to (45) are satisfied.
−Ce
−Cm
−Ce
The formulas (43) to (45) can be rearranged to be expressed as the following formulas (46) to (48).
−Cm
−Cm
−Ce
According to the following formula (49), the formula (31) is expressed as the following formula (50), and the extended digit satisfying the formula (51) is just to be determined.
C′=Cm
a1b1−C′<A×B<a1b1+C′ (50)
C′≤Cr+r′ (51)
The case of the uncertain number being 0 is the same as the case of (1) A>0 and B>0.
(3) If A<0 and B>0, then −a1≤Cma+ma′, b1≤Cmb+mb′, and the following formulas (52) to (54) are satisfied.
−Ce
−Cm
−Ce
The formulas (52) to (54) can be rearranged to be expressed as the following formulas (55) to (57).
−Cm
−Cm
−Ce
According to the following formula (58), the formula (31) is expressed as the following formula (59), and the extended digit satisfying the formula (60) is just to be determined.
C′=Cm
a1b1−C′<A×B<a1b1+C′ (59)
C′≤Cr+r′ (60)
The case of the uncertain number being 0 is the same as the case of (1) A>0 and B>0.
(4) If A<0 and B<0, then a1≤Cma+ma′, −b1≤Cmb+mb′, and the following formulas (61) to (63) are satisfied.
−Ce
−Cm
−Ce
The formulas (61) to (63) can be rearranged to be expressed as the following formulas (64) to (66).
−Cm
−Cm
−Ce
According to the following formula (67), the formula (31) is expressed as the following formula (68), and the extended digit satisfying the formula (69) is just to be determined.
C′=Cm
a1b1−C′<A×B≤a1b1+C′ (68)
C′<Cr+r′ (69)
The case of the uncertain number being 0 is the same as the case of (1) A>0 and B>0.
Division
Assume that a real number A and a real number B are the high-precision computer numbers defined by the formulas (11) and (12), respectively.
A=a1+a2,|a1|≤Ch
B=b1+b2,|b1|≤Ch
Herein a1 and b1 are definite numbers whose numerical values are definite, and a2 and b2 are uncertain numbers whose numerical values are uncertain; C=2; ha+ha and hb+hb′ respectively denote extended high order maxes that are minimum extended digits satisfying |a1|≤2ha+ha′ and |b1|≤2hb+hb′; ha and hb respectively denote high order maxes that are integers, ha′ and hb′ respectively denote decimal digits that are ha′/256 and hb′/256; ea+ea′ and eb+eb′ denote extended absolute effective digits; ea and eb denote absolute effective digits that are integers; and ea′ and eb′ respectively denote decimal digits which are decimals of ea′/256 and eb′/256.
For B/A, ha+ha′−1/256 and hb+hb′−1/256 are used instead of the extended high order max (ha+ha′, hb+hb′), and the high-precision computer number is defined by the following formulas (71) and (72).
B/A is expressed by the following formula (73).
The formula (73) is evaluated.
This formula (74) assumes that Cha+ha′-1/256−Cea+ea′ will not be 0. A value less than or equal to 0 means a possibility of A being 0, so division does not hold.
Determining the extended digit (Ha+Ha′) satisfying the formula (75) results in the following formula (76). Ha+Ha′ can be determined by referring to an extended digit subtraction table which outputs the maximum extended digit (γ+γ′) satisfying Cα′−Cβ+β′≥Cγ+γ for Cα′ and Cβ+β′.
Next, b2a1−b1a2 is evaluated for each terms. If a1>0, then the following formula (77) holds; if a1<0, then the following formula (78) holds; if b1>0, then the following formula (79) holds; and if b1<0, then the following formula (80) holds.
a1>0,−Ce
a1<0,Ce
b1>0,−Ce
b1<0,Ce
Therefore, if a1>0 and b1>0, then the following formula (81) holds; if a1>0 and b1<0, then the following formula (82) holds; if a1<0 and b1>0, then the following formula (83) holds; and if a1<0 and b1<0, then the following formula (84) holds.
a>0,b1>0,−Ce
a>0,b1<0,−Ce
a<0,b1>0,−Ce
a<0,b1<0,−Ce
Ce
Determining the extended digit (Hab+Hab′) satisfying the formula (85) results in the following formula (86).
−CH
Therefore, the second term of the formula (73) can be denoted by the following formula (87).
Next, the digit of b1/a1 will be described. Since the digit of b1/a1 can be rounded to the difference from which the extended digit (Ha+Ha′) satisfying the formula (85) is obtained in the arithmetic operation of the uncertain part, the digits up to the rounding error is just to be determined. The rounding will be described using the following formulas (88) to (90).
According to the formula (90), the digits after c can be rounded.
Next, a description will be given of a process when the uncertain number is 0, that is, when ea and eb are −∞.
(1) If ea=−∞
then, in the formula (74) for evaluating the above formula (73), determining the extended digit (Ha+Ha′) satisfying the formula (75) results in the following formula (91).
Further, determining the extended digit (Hab+Hab′) satisfying the formula (85) results in the following formula (92).
(Hab+Hab′)=(ha+eb+ha′+eb′) (92)
Therefore, according to the formula (93), the inequality of the formula (88) is rewritten as the formula (94).
In the formula (94), the formula (95) and the formula (96) are used. Then, using the difference between Cα+α′ and Cβ+β′, trimming processing of the extended absolute effective digit is performed.
If eb=−∞, then the error is zero. Therefore, it is necessary to specify the effective digit from the outside.
(2) If ea≠−∞, eb=−∞,
then determining the extended digit (Hab+Hab′) satisfying the formula (85) results in the following formula (97).
(Hab+Hab′)=(hb+ea+hb′+ea′) (97)
Therefore, according to the formula (98), the inequality of the formula (88) is rewritten as the formula (99).
Then, as in the case of (1) ea=−∞, the trimming process of the extended absolute effective digit is performed by using the difference between Cα+α′ and Cβ+β′.
Extended Digit Subtraction Table
An extended digit subtraction table for outputting the maximum extended digit (γ+γ′) satisfying Cα′−Cβ+β′≥Cγ+γ′ for Cα′ and Cβ+β′ will be described. Herein, 0≥β, and α′ and β′ are α′/28 and β′/28, respectively. If 0=β, then α′≥β′. By using the extended digit subtraction table, the arithmetic operation can be accelerated, for example, in determining the extended digit (Ha+Ha′) satisfying the formula (75).
If β=0 and α′=β′, then Cα′−Cβ+β′≥C−∞. Possible values of γ are a range of single-precision integers, and the smallest value is treated as −∞. The value is “1000000000000000” in a 2 bit representation.
An example of arithmetic operation using the high-precision computer number will be described below. The radix is 2 (C=2), and extensions (decimal part) of the exponents ha′, hb′, ea′, eb′, and r′ are ha′/28 (0≤ha′<28), hb′/28 (0≤hb′<28), ea′/28 (0≤ea′<28), eb′/28 (0≤eb′<28), and r′/28 (0≤r′<28), respectively.
In the arithmetic operation examples, double precision floating-point representations were converted into high-precision computer numbers, and arithmetic operations were performed using the high-precision computer numbers. The conversion of a double precision floating-point representation into a high-precision computer number is performed by checking the sign and exponent of the double precision floating-point representation and copying the significand.
e (−57) indicates that the effective digit is valid up to −57th digit in the binary notation. This is represented as log10 2=0.301029995663981 . . . in the decimal notation, so that log10 2−57=−17.1587097528469281 . . . , which means up to −17 digits in the decimal notation are valid (see line 4 in
To the arrays a [1] and a [0], the significand (fraction) of the double precision floating-point representation is directly substituted. It should be noted that the high-precision computer number is right-aligned. Since the value of “0.1” in the decimal notation cannot be exactly expressed in a computer, it is represented as the value in line 4.
Substituting the significand denoting “0.1” in the decimal notation in double precision floating-point representation into the array of the high-precision computer number, the following array (A) is obtained. This is a concatenation of the array a [1] and the array a [0] of the high-precision computer number shown in
0000000000001100110011001100110011001100110011001100110011001101 (A)
The first 1 bit is the high order max h and can be computed from the double precision floating-point representation. The extension (decimal digit) h′ of the high order max can be determined by referring to the above-mentioned extended digit table. The first 9 bits of the array (A) are “110011001” and the following 8 bits are “10011001”. By using these 8 bits, the extension h′ of the high order max can be determined by referring to the above-mentioned “8 bit-α′ table”. Since “10011001” is 153 in the decimal notation, the 153rd value (174) of the “8 bit-α′ table” (the first value is 0th) is read.
2174/256 is calculated as 1.601792755682693353793846241591. On the other hand, the value of the first 32 bits of the array (A) is 3435973836, and by dividing this value by 231, the first bit of the array (A) can be defined as 0th digit of the binary notation. It is understood that, since this value is 1.59999999962747097015380859375, setting the first 1 digit of the array (A) to be the 0th digit of the binary notation can suppress the array (A) to 2174/256. Since the beginning of the array (A) is −4th digit in the binary notation, 0.1≤2−4+174/256 is satisfied.
When actually calculated, the following arithmetic operation results are obtained, and it is found that 2−4+173/256≤0.1≤2−4+174/256.
2−4+174/256=0.10011204723016833461211539009943
2−4+173/256=0.09984134986928919364142626838804
Addition
In the high-precision computer number shown in
267/256=1.19890616707438048177 . . .
268/256=1.2021567314527031420 . . .
The extended absolute effective digits e (−53) and e′ (23) of the arithmetic operation result indicate that the extended absolute effective digits e (−57) and e′ (0) of the high-precision computer number of “0.1” and the extended absolute effective digits e (−53) and e′ (0) of the high-precision computer number of “1.1” are used and that 2−53+22/256≤2−57+2−53≤2−53+23/256. Multiplying both sides by 253 yields the following arithmetic operation result, to reveal that 222/256≤2−4+1≤223/256.
222/256=1.061377227289262 . . .
2−4+1=1.0625
223/256=1.0642549128844645 . . .
Next, the offset value is determined.
0.1=0.10000000000000000555111512312+α|α|≤2−57
1.1=1.1000000000000000888178419700+β|β|≤2−53
1.2=1.20000000000000009436895709312+α+β
Since α+β≤2−57+2−53=2−53 (2−4+1), the minimum γ′ satisfying (1+2−4)≤2γ′/256 is determined. In (1+2−4), 1=2°, 2−4=2−4, α′=0, β=−4, and β′=0, so that γ′ and the digits of the difference (offset value) are obtained from these values by referring to the above-mentioned extended digit addition table.
Using the value (γ′=23) in the extended digit addition table, (1+2−4)≤223/256; actual calculation of both sides yields the following arithmetic operation result.
1+2−4=1.0625
223/256=1.0642549128844645497886112570016
The offset value of the extended digit addition table is 10. The arithmetic operation result of the difference is as follows.
223/256−(1+2−4)=0.00175491288446454978861125700158
The arithmetic operation result obtained by multiplying this by 210 is as follows. 1.7970307936916989835379271696181
Digits after the decimal point in the binary notation are “0.0000000001********”, and the part of “1********” or less can be truncated. This indicates that 10 digits or less from −53rd digit in the binary notation may be truncated. Thus, the low order max l is −55th digit, which does not exceed the absolute effective digit e (−53). Originally, −55th digits or less in the binary notation are 0s.
In the iterative arithmetic operation of computer numbers according to the first embodiment, since the value of the absolute effective digit e is increased by about 3 for each arithmetic operation, the significance of the effective digit is lost when the arithmetic operations are performed 100 times, but this can be improved by the high-precision computer number.
Multiplication
In the high-precision computer number shown in
2−4+208/256=0.1097657600233312 . . .
2−4+209/256=0.11006336518984898 . . .
In the high-precision computer number shown in
C′=Cm
C′=20−57+36/256+0+2−4−53+174/256+0+2−57−53+0+0
C′=2−57+36/256+2−57+174/256+2−110
257C′=236/256+2174/256+2−53
When actually calculated, the following arithmetic operation results are obtained, and it is found that 257 (2−56+111/256)≤257C′≤257 (2−56+112/256). It can be also seen that e (−56) and e′ (112) are the minimum extended digits satisfying C′≤Cγ+γ′, and are the extended effective digits (e+e′/256).
257(2−56+112/256)=2.708511093873 . . .
257C′=2.704175338990 . . .
257(2−56+111/256)=2.701187431 . . .
Since the offset value is 9, the low order max l is (−63), which is smaller than e (−56). It is assumed that two or more 0s are consecutive after this digit.
The effective digits are compared between the arithmetic operation result shown in
FIG. 33: (0+36/256)−(−53)=53+36/256
FIG. 42: (−2326+202/256)−(−2372+1/256)=43+203/256
Although the effective digit is reduced by about 6+89/256, they are improved compared with the computer numbers in the first embodiment.
Division
In the above formula (75), ha+ha′−1/256, ea+ea′ are calculated as follows.
ha+ha′−1/256=−4+173/256
ea+ea′−57
Then, these numerical values are used to calculate Ha+Ha by referring to the extended digit subtraction table. Because ea is smaller than ha, the value is obtained by subtracting 1/256 from ha+ha−1/256.
Ha+Ha′=−4+172/256
In the above formula (99), (hb+ea−Ha−ha)+(hb′+ea′−Ha′−ha′+1/256) is −51+168/256, which is the arithmetic operation result of the extended absolute effective digits e (−51) and e′ (169). Since the offset value is 8, the extended absolute effective digit is larger than the low order max l (−53).
In the above formula (75), ha+ha′−1/256 and ea+ea′ are calculated as follows.
ha+ha′−1/256=−4+173/256
ea+ea′=−57
Then, these numerical values are used to calculate Ha+Ha′ by referring to the extended digit subtraction table. Because ea is smaller than ha, the value is obtained by subtracting 1/256 from ha+ha′−1/256.
Ha+Ha′=−4+172/256
In the above formula (89), the following two values of the exponent are calculated.
(eb−Ha)+(eb′−Ha′+1/256)=−53+85/256
(hb+ea−Ha−ha)+(hb′+ea′−Ha′−ha′+1/256)=−53+85/256
Since there is no difference between these two values, in the method in which the offset value is ∞, the extended absolute effective digit of the arithmetic operation result is −52+85/256 plus 1/256 (e (−52), e′ (86)).
11 input unit, 12 storage unit, 13 arithmetic unit, 14 control unit, 21 CPU, 22 GPU, 23 ROM, 24 RAM, 25 operation input unit, 26 storage, 27 input/output interface, 31 input unit, 32 storage unit, 33 arithmetic unit, 34 control unit
Number | Date | Country | Kind |
---|---|---|---|
JP2019-214678 | Nov 2019 | JP | national |
Filing Document | Filing Date | Country | Kind |
---|---|---|---|
PCT/JP2020/024260 | 6/19/2020 | WO | 00 |
Publishing Document | Publishing Date | Country | Kind |
---|---|---|---|
WO2021/106253 | 6/3/2021 | WO | A |
Number | Name | Date | Kind |
---|---|---|---|
20020019975 | Johnson | Feb 2002 | A1 |
20140074902 | Naeimi | Mar 2014 | A1 |
Number | Date | Country |
---|---|---|
2004-005395 | Jan 2004 | JP |
2006-323710 | Nov 2006 | JP |
2015-232833 | Dec 2015 | JP |
2017-538197 | Dec 2017 | JP |
Entry |
---|
Jul. 28, 2020 Written Opinion of the International Searching Authority issued in International Patent Application No. PCT/JP2020/024260. |
Number | Date | Country | |
---|---|---|---|
20210247960 A1 | Aug 2021 | US |