```Issue #8700 has been updated by matz (Yukihiro Matsumoto).

Accepted.  It should be work as 2's complement for negative numbers.

Matz.

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Feature #8700: Integer#bitsize (actually Fixnum#bitsize and Bignum#bitsize)
https://bugs.ruby-lang.org/issues/8700#change-41474

Author: akr (Akira Tanaka)
Status: Open
Priority: Normal
Assignee:
Category:
Target version:

Integer#bitsize returns the position of the most significant bit in the absolute value.
(The position of the least significant bit is 1.)
It returns 0 if no bit set (i.e. the value 0).

Mathematically, n.bitsize is ceil(log2(abs(n)+1)).

Sometimes we want to know the size of a integer.

* Determine the size of an integer in some format.
Although there are various formats, bitsize is a key property to determine the result size.
Several examples:
* If a format is 4 bytes for absolute value, it overflows if 32 <= n.bitsize.
* If a format is 4 bytes for sign bit with absolute value, it overflows if 31 <= n.bitsize.
* If a format is 4 bytes for 2's complement format, it overflow if 31 <= n.bitsize && n != -2**31.
* BER-compressed integer needs (n.bitsize+6)/7 bytes when n > 0.
BER-compressed integer is an example of VLQ.
http://en.wikipedia.org/wiki/Variable-length_quantity
* Elias gamma coding needs 2*n.bitsize-1 bits.
https://en.wikipedia.org/wiki/Elias_gamma_coding
* Elias delta coding needs 2*n.bitsize.bitsize+n.bitsize-2 bits.
https://en.wikipedia.org/wiki/Elias_delta_coding

* bitsize may be used to estimate the time or space cost of an algorithm.
For example, the result size of integer multiplication, x*y, is x.bitsize + y.bitsize.
The number of comparisons of binary search is sorted_array.length.bitsize, etc.
This is because n.bitsize is an approximation of log2(abs(n)).
So Math.log2 can be used for this purpose too.
However bitsize may be preferable if floating point error is not desirable.

There are several software which has similar feature.

* Python 3.1 has int.bit_length().
http://docs.python.org/dev/library/stdtypes.html
http://docs.python.org/3.1/whatsnew/3.1.html
http://bugs.python.org/issue3439

* Java java.math.BigInteger has bitLength() method.
http://docs.oracle.com/javase/7/docs/api/java/math/BigInteger.html#bitLength()

* Mathematica has BitLength.
http://reference.wolfram.com/mathematica/ref/BitLength.html

* GMP has mpz_sizeinbase(n, base).
http://gmplib.org/manual/Miscellaneous-Integer-Functions.html

* NetBSD 5.0 has ilog2().
http://netbsd.gw.com/cgi-bin/man-cgi?ilog2++NetBSD-6.0

I think there are two concerns for this issue.
* method name
* behavior for zero and negative number

I named the method as bitsize, mainly because
there is Fixnum#size and Bignum#size.
However I'm open for other names such as:
* bitlength
* numbits
* ilog2
* maxbit
Some names may suggest different behavior, though.

The behavior for zero and negative number is not trivial.

Java and Mathematica adopts ceil(log2(n < 0 ? -n : n+1)).
The difference is absolute number v.s. 2's complement number.

Some people may prefer ilog2, which name suggests ilog2(0) raise an error.

I choose ceil(log2(abs(n)+1)). (i.e. absolute number, same as Python).
I think absolute number is easier to understand than 2's complement for many people.

I attached the implementation as bitsize.patch.
The patch implements both Bignum#bitsize and Fixnum#bitsize in bignum.c.
It is because Fixnum#bitsize uses bitsize macro and it is defined in bignum.c.
Maybe, the macro should be moved to internal.h and the implementation of
Fixnum#bitsize should be moved to numeric.c.