Computer implemented systems and methods of addition of numbers represented in a logarithmic number system

A computer system comprising:

• a memory;
• and
• one or more processors in communication with the memory, wherein the computer system is configured to perform a method for performing addition of real numbers represented in a multidimensional logarithmic number system (MDLNS), the method comprising:
• representing two real numbers X and Y in an MDLNS, wherein the MDLNS representations of X and Y includes at least two bases B1 and B2, as follows:
• X
• =
• s
• ⁢
• 1
• *
• (
• B
• ⁢
• 1
• ^
• a
• )
• *
• (
• B
• ⁢
• 2
• ^
• b
• )
• ,
• and
• Y
• =
• s
• ⁢
• 2
• *
• (
• B
• ⁢
• 1
• ^
• c
• )
• *
• (
• B
• ⁢
• 2
• ^
• d
• )
• ,
• wherein s1 and s2 are each a sign of value plus or minus one, and wherein a, b, c and d are exponents,
• imposing conditions on the exponents as follows:
• a, b, c and d are integers, and
• a is positive, b is negative, c is negative and d is positive, and
• expanding the representation of the sum of X+Y in the log domain as follows:
• log (X+Y) = log (s1*(B1{circumflex over ( )}a)*(B2{circumflex over ( )}b) + s2*(B1{circumflex over ( )}c)*(B2{circumflex over ( )}d)) =
• log (s1*(B1{circumflex over ( )}a*B2{circumflex over ( )}d)*(B2{circumflex over ( )}(b−d) + (s2/s1)*B1{circumflex over ( )}(c−a))), wherein (b−d) is a
• negative exponent and (c−a) is a negative exponent.

A computer program product comprising:

• one or more computer readable storage media and program instructions collectively stored on the one or more computer readable storage media readable by at least one processing circuit, wherein the computer program product is configured to perform a method for performing addition of real numbers represented in a multidimensional logarithmic number system (MDLNS), the method comprising:
• representing two real numbers X and Y in a multidimensional logarithmic number system (MDLNS), wherein the MDLNS representations of X and Y includes at least two bases B1 and B2, as follows:
• X
• =
• s
• ⁢
• 1
• *
• (
• B
• ⁢
• 1
• ^
• a
• )
• *
• (
• B
• ⁢
• 2
• ^
• b
• )
• ,
• and
• Y
• =
• s
• ⁢
• 2
• *
• (
• B
• ⁢
• 1
• ^
• c
• )
• *
• (
• B
• ⁢
• 2
• ^
• d
• )
• ,
• wherein s1 and s2 are each a sign of value plus or minus one, and wherein a, b, c and d are exponents;
• imposing conditions on the exponents as follows:
• a, b, c and d are integers, and
• a is positive, b is negative, c is negative and d is positive;
• and
• expanding the representation of the sum of X+Y in the log domain as follows:
• log (X+Y) = log (s1*(B1{circumflex over ( )}a)*(B2{circumflex over ( )}b) + s2*(B1{circumflex over ( )}c)*(B2{circumflex over ( )}d)) =
• log (s1*(B1{circumflex over ( )}a*B2{circumflex over ( )}d)*(B2{circumflex over ( )}(b−d) + (s2/s1)*B1{circumflex over ( )}(c−a))), wherein (b−d) is a negative
• exponent and (c−a) is a negative exponent.

A computer-implemented method for performing addition of real numbers represented in a multidimensional logarithmic number system (MDLNS), the method comprising:

• obtaining, by program code executing on one or more processors, an instruction to sum together a first real number and a second real number;
• representing, by the program code, the first and second real numbers an MDLNS, wherein the MDLNS representation of the first and second real numbers includes:
• the first real number represented by a first base raised to a first exponent multiplied by a second base raised to a second exponent, and
• the second real number represented by the first base raised to a third exponent multiplied by the second base raised to a fourth exponent;
• imposing conditions, by the program code, on the exponents, wherein:
• the exponents are integers, and
• the first exponent is positive, the second exponent is negative, the third exponent is negative and the fourth exponent is positive;
• expanding, by the program code, an MDLNS representation of a sum of the first and second real numbers in a log domain;
• approximating, by the program code, a value of a portion of the expanded MDLNS representation of the sum in a form of an approximated product term;
• and
• utilizing, by the program code, the approximated product term to simplify the MDLNS representation of the sum of the first and second real numbers, wherein the MDLNS representation of the sum is approximated as the first base raised to a fifth integer exponent multiplied by the second base raised to a sixth integer exponent, and wherein the actual value of the sum and the approximated value of the sum are within at least 4 bits of precision or within at least 6.4 percent.