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inverse square root

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i've read this ( http://www.math.purdue.edu/~clomont/Math/Papers/2003/InvSqrt.pdf ) article a few times now but a still confuses fuses me could someone please explain it to me?

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"The true exponent e = E − 127 is odd, say e = 2d + 1, d an integer.
The bits representing the initial guess give the new exponent"

after that i seem to start getting confused with what hes doing.

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Could you write down a representation of a floating point number in the registry? That article was somehow harder to read. He wrote it in scientifical style that gets in the way.

Try to do some example calculation, for example 1/sqrt(2) or 1/sqrt(4)

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a = 2*b + 1 guarantees the "a" would be odd. (b is an integer)

He just used a "d" instead of a "b".

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