I want to convert the number 8.7 to binary.
I know the commands
(format nil "~b" (rationalize 8.7)) ===>
1010111/1010
or
(format nil "~b" (/ 87 10))====> 1010111/1010
We observe if we do the quotient binary 1010111/1010 we obtain 1000.1011001100110011001100110011.
Is possible to obtain in Lisp
(8.7)_2 ~ 1000.1011001100110011001100110011?
If yes, how?
"2.718..." is equal to 2 * 10^1 + 7 * 10^-1 + 1 * 10^-2 + 8 * 10^-3... This means that you can generate the string by the reverse process of concatenating string(n / base^i) where i is the index into the string and n is the value that still needs to be converted to base. It's essentially a greedy change-making algorithm.
The following roughly works, with no guarantee that it produces the exact IEEE 754 fraction. It should be as accurate as your implementation's floats
(defun fractional-binary (fractional &optional (place 1) (result "."))
(if (zerop fractional)
result
(let ((fraction (expt 2 (- place))))
(if (<= fraction fractional)
(fractional-binary (- fractional fraction)
(1+ place)
(concatenate 'string result "1"))
(fractional-binary fractional
(1+ place)
(concatenate 'string result "0"))))))
CL-USER> (fractional-binary .7)
".101100110011001100110011"
