Information processing apparatus, program, and information processing method configured to handle a high-precision computer number

A non-transitory recording medium comprising a program for causing a computer to execute a process comprising:

• 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 bit representation into a memory;
• and
• an arithmetic step of performing an arithmetic operation by using definite numbers a1 and b1 of the high-precision computer numbers stored in the memory, and by assuming that real numbers A and B fall within a range of a1−Cea+ea′A≤A<a1+Cea+ea′ and b1−Ceb+eb′, respectively, determining an extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13);
• and storing, in the memory, high-precision computer numbers as the result of the arithmetic operation,
• A=a1+a2,|a1|≤Cha+ha′,−Cea+ea′≤a2<Cea+ea′ (11)
• B=b1+b2,|b1|≤Chb+hb′,−Ceb+eb′≤b2<Ceb+eb′ (12)
• (A,B)=(a,b)1+(a,b)2,−Cγ+γ′≤(a,b)2<Cγ+γ′ (13)
• 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 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;
• and
• wherein a sign s, a high order max h, a high order max sub h′, a low order max l, an absolute effective digit e, an absolute effective digit sub e′, and a number of array a are input, a predetermined number of bit arrays in the memory are secured based on the high order max h and a least significant digit l, and the high-precision computer number is stored in the memory in bit representation.

An information processing method comprising:

• 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 bit representation into a memory;
• and
• an arithmetic step of performing an arithmetic operation by using definite numbers a1 and b1 of the high-precision computer numbers stored in the memory, and by assuming that real numbers A and B fall within a range of a1−Cea+ea′A≤A<a1+Cea+ea′ and b1−Ceb+eb′, respectively, determining an extended absolute effective digit (γ+γ′) so as to satisfy the following formula (13);
• and storing, in the memory, high-precision computer numbers as the result of the arithmetic operation,
• A=a1+a2,|a1|≤Cha+ha′,−Cea+ea′≤a2<Cea+ea′ (11)
• B=b1+b2,|b1|≤Chb+hb′,−Ceb+eb′≤b2<Ceb+eb′ (12)
• (A,B)=(a,b)1+(a,b)2,−Cγ+γ′≤(a,b)2<Cγ+γ′ (13)
• 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 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;
• and
• wherein a sign s, a high order max h, a high order max sub h′, a low order max l, an absolute effective digit e, an absolute effective digit sub e′, and a number of array a are input, a predetermined number of bit arrays in the memory are secured based on the high order max h and a least significant digit l, and the high-precision computer number is stored in the memory in bit representation.