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Add documentation note about signed overflow direction In https://github.com/rust-lang/rust/issues/151989#issuecomment-3845282666 I noticed that signed overflow direction can be determined by returned wrapped value. It is not very obvious (especially, assuming additional `carry: bool` summand), but it is important if we want to add new leading (signed) limb to big integer in this case. Examples for small summands `x, y: i8` with result extension: | x | y | overflow | result as (u8, i8) | | ---- | ---- | -------- | ------------------ | | -1 | -128 | true | (127, -1) | | 0 | -1 | false | (255, -1) | | 2 | 2 | false | (4, 0) | | 127 | 1 | true | (128, 0) | Here is general proof. 1. Set $s=2^{N-1}$ and let's say `iN::carrying_add(x, y, c)` returns `(result, true)` then $$ \mathrm{result}=\begin{cases} x + y + c + 2s,& x + y + c \le -s-1,\\ x+y+c-2s,& x+y+c\ge s. \end{cases} $$ First case is overflowing below `iN::MIN` and we have $$ \mathrm{result}\ge -s-s+0+2s =0;\qquad \mathrm{result}=x + y + c + 2s\le -s-1+2s = s - 1, $$ so we obtain $[0; s-1]$ which is exactly range of non-negative `iN`. Second case is overflowing above `iN::MAX` and $$ \mathrm{result}=x+y+c-2s\ge s-2s =-s;\qquad \mathrm{result}\le s-1 + s-1+1-2s = -1, $$ that is, $[-s,-1]$ which is exactly range of negative `iN`. 2. Now suppose that `iN::borrowing_sub(x,y,b)` returns `(result, true)` then $$ \mathrm{result}=\begin{cases} x - y - b + 2s,& x - y - b \le -s-1,\\ x - y - b - 2s,& x - y - b\ge s. \end{cases} $$ First case is overflowing below `iN::MIN` and we have $$ \mathrm{result}\ge -s-(s-1)-1+2s =0;\qquad \mathrm{result}=x - y - b + 2s\le -s-1+2s = s - 1. $$ Second case is overflowing above `iN::MAX` and $$ \mathrm{result}=x-y-b-2s\ge s-2s =-s;\qquad \mathrm{result}\le s-1 - (-s) - 0 - 2s = -1. $$ |
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