BKM algorithm

In order to solve the equation

${\displaystyle \ln(x)=y}$ 

the BKM algorithm takes advantage of a basic property of logarithms

${\displaystyle \ln(ab)=\ln(a)+\ln(b)}$ 

Using Pi notation, this identity generalizes to

${\displaystyle \ln \left(\prod _{k=0}^{n}a_{k}\right)=\sum _{k=0}^{n}\ln(a_{k})}$ 

Because any number can be represented by a product, this allows us to choose any set of values ${\displaystyle a_{k}}$  which multiply to give the value we started with. In computer systems, it's much faster to multiply and divide by multiples of 2, but because not every number is a multiple of 2, using ${\displaystyle a_{k}=1+2^{m}}$  is a better option than a more simple choice of ${\displaystyle a_{k}=2^{m}}$ . Since we want to start with large changes and get more accurate as ${\displaystyle k}$  increases, we can more specifically use ${\displaystyle a_{k}=1+2^{-k}}$ , allowing the product to approach any value between 1 and ~4.768, depending on which subset of ${\displaystyle a_{k}}$  we use in the final product. At this point, the above equation looks like this:

${\displaystyle \ln \left(\prod _{k\in K\subset \mathbb {Z} _{0}^{+}}1+2^{-k}\right)=\sum _{k\in K\subset \mathbb {Z} _{0}^{+}}\ln(1+2^{-k})}$ 

This choice of ${\displaystyle a_{k}}$  reduces the computational complexity of the product from repeated multiplication to simple addition and bit-shifting depending on the implementation. Finally, by storing the values ${\displaystyle \ln(1+2^{-k})}$  in a table, calculating the solution is also a simple matter of addition. Iteratively, this gives us two separate sequences. One sequence approaches the input value ${\displaystyle x}$  while the other approaches the output value ${\displaystyle \ln(x)=y}$ :

$${\displaystyle x_{k}={\begin{cases}1&{\text{if }}k=0\\x_{k-1}\cdot (1+2^{-k})&{\text{if }}x_{k}{\text{ would be}}\leq x\\x_{k-1}&{\text{otherwise}}\end{cases}}}$$ 

Given this recursive definition and because ${\displaystyle x_{k}}$  is strictly increasing, it can be shown by induction and convergence that

${\displaystyle \lim _{k\to \infty }x_{k}=x}$ 

for any ${\displaystyle 1\leq x\lesssim 4.768}$ . For calculating the output, we first create the reference table

${\displaystyle A_{k}=\ln(1+2^{-k})}$ 

Then the output is computed iteratively by the definition $${\displaystyle y_{k}={\begin{cases}0&{\text{if }}k=0\\y_{k-1}+A_{k}&{\text{if }}x_{k}{\text{ would be}}\leq x\\y_{k-1}&{\text{otherwise}}\end{cases}}}$$  The conditions in this iteration are the same as the conditions for the input. Similar to the input, this sequence is also strictly increasing, so it can be shown that

${\displaystyle \lim _{k\to \infty }y_{k}=y}$ 

for any ${\displaystyle 0\leq y\lesssim 1.562}$ .

Because the algorithm above calculates both the input and output simultaneously, it's possible to modify it slightly so that ${\displaystyle y}$  is the known value and ${\displaystyle x}$  is the value we want to calculate, thereby calculating the exponential instead of the logarithm. Since x becomes an unknown in this case, the conditional changes from

${\displaystyle \dots {\text{if }}x_{k}{\text{ would be}}\leq x}$ 

to

${\displaystyle \dots {\text{if }}y_{k}{\text{ would be}}\leq y}$ 

To calculate the logarithm function (L-mode), the algorithm in each iteration tests if ${\displaystyle x_{n}\cdot (1+2^{-n})\leq x}$ . If so, it calculates ${\displaystyle x_{n+1}}$  and ${\displaystyle y_{n+1}}$ . After ${\displaystyle N}$  iterations the value of the function is known with an error of ${\displaystyle \Delta \ln(x)\leq 2^{-N}}$ .

Example program for natural logarithm in C++ (see A_e for table):

 double log_e (const double Argument, const int Bits = 53) // 1 <= Argument <= 4.768462058
 {
    double  x = 1.0, y = 0.0, s = 1.0;

    for (int k = 0; k < Bits; k++) {
      double const  z = x + x*s;
      if (z <= Argument) {
          x  = z;
          y += A_e[k];
      }
      s *= 0.5;
    }
    return y;
 }

Logarithms for bases other than e can be calculated with similar effort.

Example program for binary logarithm in C++ (see A_2 for table):

 double log_2 (const double Argument, const int Bits = 53) // 1 <= Argument <= 4.768462058
 {
    double  x = 1.0, y = 0.0, s = 1.0;

    for (int k = 0; k < Bits; k++) {
      double const  z = x + x*s;
      if (z <= Argument) {
          x  = z;
          y += A_2[k];
      }
      s *= 0.5;
    }
    return y;
 }

The allowed argument range is the same for both examples (1 ≤ Argument ≤ 4.768462058…). In the case of the base-2 logarithm the exponent can be split off in advance (to get the integer part) so that the algorithm can be applied to the remainder (between 1 and 2). Since the argument is smaller than 2.384231…, the iteration of k can start with 1. Working in either base, the multiplication by s can be replaced with direct modification of the floating point exponent, subtracting 1 from it during each iteration. This results in the algorithm using only addition and no multiplication.

To calculate the exponential function (E-mode), the algorithm in each iteration tests if ${\displaystyle y_{n}+\ln(1+2^{-n})\leq y}$ . If so, it calculates ${\displaystyle x_{n+1}}$  and ${\displaystyle y_{n+1}}$ . After ${\displaystyle N}$  iterations the value of the function is known with an error of ${\displaystyle \Delta \exp(x)\leq 2^{-N}}$ .

Example program in C++ (see A_e for table):

 double exp (const double Argument, const int Bits = 54)	// 0 <= Argument <= 1.5620238332
 {
   double  x = 1.0, y = 0.0, s = 1.0;

   for (int k = 0; k < Bits; k++) {
      double const  z = y + A_e[k];
      if (z <= Argument) {
         y = z;
         x = x + x*s;
      }
      s *= 0.5;
   }
   return x;
 }

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