Background of my reference request is an observation that I made, while I was still in school: there are two ways to calculate $x*999$: either do it directly, by applying the multiplication algorithm that is taught in school or, calculate it as $x*1000-x*1$ of which the second way is much easier.
But, I had no clue about the rules behind such a simplification.

Later, having studied computer science and math, and having learned of binary digits, things got clearer and I realized that the objective would be to express the multiplicator as a difference of non-negative integers with a minimal number of 1-bits in the binary representation.
Despite the fact that such a representation could speed up multiplication via repeated addition plus one subtraction or, exponentiation via repeated squaring plus one division, I could not find any mentioning of that representation or how to obtain and optimal such representation.

I would therefore appreciate any pointers to information about the construction and properties of the representation of natural numbers as a difference with minimal Hamming weight.

My currently best algorithm to determine a such difference-representation with a small Hamming weight is to
first fill "fissions", i.e. replace 0-bits that are next to two 1-bits on the left and on the right, so $...11011...$ becomes $...11111...$
and then replace each bit of an uninterrupted sequences of at least three 1-bits by 0-bits and, each 0-bit that is immediately to the left of such a sequence, by 1-bit so that $...011...10...$ would become $...100...00...$
With that operations, 27 would first become 31 due to fixing the fission and finally 32; the difference-representation is then 32-5 with a Hamming weight of 3 instead of 4.

An interesting phenomen is, that there are bit-patterns like $0010011011$, for which the difference encoding reduces the Hamming weight also for the bit-complement and, there are others, like $0110011$, for which no improvement is possible, even for the bit-complement.
This raises the question about the statistical properties of the quotient of the Hamming weights of difference encoding and of standard encoding of numbers.

In order to give some 4impression of the amount of operations that can be saved, I applied the method to the firt 200 binary digits of some wellknown constants; the percentages relate to the number of 1-bits:

$60\% \approx 67/111$ Khinchin constant
$66\% \approx 71/108$ Chapernowne constant
$66\% \approx 72/109$ ln(2)
$68\% \approx 75/111$ Conway constant
$68\% \approx 75/110$ sqrt(2)
$69\% \approx 69/100$ Euler-Mascheroni constant
$70\% \approx 65/093$ Apéry constant
$73\% \approx 74/101$ Plastic Number
$73\% \approx 77/105$ $e$
$73\% \approx 77/105$ Golden ratio
$79\% \approx 70/089$ Feigenbaum constant
$83\% \approx 68/082$ $\pi$

From this results, it seems unlikely that the method has not been described previously because of only negligible savings.