2LOG, calculate binary logarithm
The 2LOG routine was the result of the need for converting a linear input into something with a more logarithmic character. There was no need for high accuracy, but rather a short reliable routine. The routine uses a very simple principle, but the result is surprisingly useable. A prime example where focussing on the essentials results in an excellent solution.
The 2LOG routine produces as output a fixed floating point number with 8 bits after the decimal point. ( x…x.xxxxxxxx ) The number of bits before the decimal point depends on the native width of the stack.
The principle used is this:
looking at the number to be converted in binary representation: (see grafic below) - step 1: take the bit-position+1 of the first set bit as number before the decimal point - step 2: as 8 bit fraction take the highest 8 bits following the most most significant set bit. If there are less than 8 bits, pad the end with cleared bits up to 8 bits.
It is good to notice that the fractional part forms a linear interpolation between two consecutive log numbers. For most purposes that is accurate enough.
the generic Forth program
As example we look at the routine for a 16b Forth. The other example is suitable for all Forth implementations.
decimal : 2LOG16b ( u -- y ) 16 0 do s>d if 2* 8 rshift \ linear interpolation 15 i - \ logarithmic class 8 lshift or leave then 2* loop ;
- The programs does maximal 16 loops.
- For each loop S>D is used to check if the most significant bit is set. If not, then the program shifts the number 1 bit to the left with 2*.
- Otherwise it calculates the output.
- The logarithmic part is calculated by subtracting the index from 15.
- The fractional part is calculated by shifting to left with 1bit, followed by shifting 8 bits to the right.
- Finally, both numbers are then combined into one number with a shift and or as final output.
The general version
The general version is suitable for all Forth-implementations which have a multiple of 8 bits as cell-width. It functions in exactly the same way as the 16b example above. But during compilation it calculates the, for that Forth-implementation relevant, values for the do…loop, shift and subtraction.
: 2LOG ( u -- y ) [ 8 cells ] literal 0 do \ #bits/cell s>d if 2* [ 8 cells 8 - ] literal rshift \ linear interpolation [ 8 cells 1- ] literal i - \ logarithmic class 8 lshift or leave then 2* loop ;