Editing Binary logarithm (section)

===Piecewise-linear approximation===
For a number <math>x</math> represented in [[floating point]] as <math>x=2^E(1+m)</math>, with integer exponent <math>E</math> and [[significand|mantissa]] <math>m</math> in the range <math>0\le m<1</math>, the binary logarithm can be roughly approximated as <math>\log_2 x\approx E+m</math>.<ref name=mitchell/> This approximation is exact at both ends of the range of mantissas but underestimates the logarithm in the middle of the range, reaching a maximum error of approximately 0.086 at a mantissa of approximately 0.44. It can be made more accurate by using a [[piecewise linear function]] of <math>m</math>,<ref>{{citation
 | last1 = Combet | first1 = M.
 | last2 = Van Zonneveld | first2 = H.
 | last3 = Verbeek | first3 = L.
 | date = December 1965
 | doi = 10.1109/pgec.1965.264080
 | issue = 6
 | journal = IEEE Transactions on Electronic Computers
 | pages = 863–867
 | title = Computation of the base two logarithm of binary numbers
 | volume = EC-14}}</ref> or more crudely by adding a constant correction term <math>\log_2 x\approx E+m+\sigma</math>. For instance choosing <math>\sigma\approx 0.043</math> would halve the maximum error. The [[fast inverse square root]] algorithm uses this idea, with a different correction term that can be inferred to be <math>\sigma\approx 0.0450466</math>, by directly manipulating the binary representation of <math>x</math> to multiply this approximate logarithm by <math>-\tfrac12</math>, obtaining a floating point value that approximates <math>1/\sqrt x</math>.<ref>{{citation
|url=http://www.daxia.com/bibis/upload/406Fast_Inverse_Square_Root.pdf
|title=The Mathematics Behind the Fast Inverse Square Root Function Code
|last=McEniry
|first=Charles
|date=August 2007
|archive-url=https://web.archive.org/web/20150511044204/http://www.daxia.com/bibis/upload/406Fast_Inverse_Square_Root.pdf
|archive-date=2015-05-11}}</ref>