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

A computer-implemented method of subtraction of real numbers represented in a parallel adaptive logarithmic number system (PALNS), the method comprising:

• representing two positive real numbers x and y in a PALNS, as follows:
• x
• =
• (
• B
• ⁢
• 1
• ^
• a
• )
• *
• (
• B
• ⁢
• 2
• ^
• b
• )
• ,
• y
• =
• (
• B
• ⁢
• 1
• ^
• c
• )
• *
• (
• B
• ⁢
• 2
• ^
• d
• )
• ,
• wherein B1 and B2 are bases, and
• wherein x≥y, and a, b, c and d are positive integer exponents;
• calculating a value R=LogB1B2;
• estimating R≈R1/R2, wherein R1 and R2 are positive integers and R1≥R2;
• designating a discriminator K=(R2*(a−c)+R1*(b−d);
• determining the best approximation of B1{circumflex over ( )}(K/R2)−1 as (B1{circumflex over ( )}g(K))*(B2{circumflex over ( )}h(K)) for every integer of a plurality of integers of K, wherein g(K) and h(K) are restricted to integers as a function of the value of K;
• and
• estimating the result of x−y, for every integer of the plurality of integers of K, as:
• x−y≈B1{circumflex over ( )}(c+g(K))*B2{circumflex over ( )}(d+h(K))=(B1{circumflex over ( )}e(K))*(B2{circumflex over ( )}f(K)) wherein e(K)=c+g(K) and f(K)=d+h(K).

A computer system comprising:

• a memory;
• a processor in communication with the memory;
• and
• an algorithm stored in the memory as a set of instructions for execution by the processor, the algorithm programmed to perform a method of addition of real numbers represented in a parallel adaptive logarithmic number system (PALNS), the method comprising:
• representing two positive real numbers x and y in a PALNS, as follows:
• x=(B1{circumflex over ( )}a)*(B2{circumflex over ( )}b),
• y=(B1{circumflex over ( )}c)*(B2{circumflex over ( )}d),
• wherein B1 and B2 are bases, and
• wherein y≥x, and a, b, c and d are positive integer exponents,
• calculating a value R=LogB1B2,
• estimating R≈R1/R2, wherein R1 and R2 are positive integers and R1≥R1,
• designating a discriminator K=(R2*(c−a)+R1*(d−b),
• determining the best approximation of 1+B1{circumflex over ( )}(K/R2) as (B1{circumflex over ( )}g(K))*(B2{circumflex over ( )}h(K)) for every integer of a plurality of integers of K, wherein a (K) and b (K) are restricted to integers as a function of the value of K, and
• estimating the sum of x+y, for every integer of the plurality of integers of K, as:
• x+y≈B1{circumflex over ( )}(a+g(K))*B2{circumflex over ( )}(b+h(K))=(B1{circumflex over ( )}e(K))*(B2{circumflex over ( )}f(K)), wherein e(K)=a+g(K) and f(K)=b+h(K).