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# paniq/minmaxabssign.txt

Last active Dec 23, 2021
useful min/max/abs/sign identities
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 max(-x,-y) = -min(x,y) min(-x,-y) = -max(x,y) abs(x) = abs(-x) abs(x) = max(x,-x) = -min(x,-x) abs(x*a) = if (a >= 0) abs(x)*a (a < 0) -abs(x)*a // basically any commutative operation min(x,y) + max(x,y) = x + y min(x,y) ^ max(x,y) = x ^ y min(x,y) * max(x,y) = x * y min(x,y) = (x+y-abs(x-y))/2 max(x,y) = (x+y+abs(x-y))/2 min(x+a,y+a) = min(x,y)+a max(x+a,y+a) = max(x,y)+a min(x*a,y*a) = if (a >= 0) min(x,y)*a (a < 0) max(x,y)*a max(x*a,y*a) = if (a >= 0) max(x,y)*a (a < 0) min(x,y)*a max(x,0)^2 = x*max(x,0) x = sign(x) * abs(x) abs(x) = sign(x) * x x/abs(x) = abs(x)/x sign(x) = floor(x / (abs(x)+1)) - floor(-x / (abs(x)+1)) (without zero:) sign(x) = floor(x / (abs(x)+1))*2+1 sign(x) ~= x/sqrt(x^2 + E^2), E -> 0 abs(x-y) <= abs(x-C) + abs(C-y) abs(x+y) <= abs(x) + abs(y) abs(x-y) >= abs(abs(x) - abs(y)) abs(x*y) = abs(x)*abs(y) max(abs(x),abs(y))*2 = abs(x) + abs(y) + abs(abs(x) - abs(y)) = abs(x + y) + abs(x - y) a + abs(b) = max(a + b, a - b) # assuming a,b,x in [1,-1] true = 1 false = -1 not x = -x a and b = (1+a)*(1+b)/2 - 1 a or b = 1 - (1-a)*(1-b)/2 a xor b = -a*b a > b = sign(a - b) # assuming sign(0) == 1 a < b = sign(b - a) # assuming sign(0) == 1 u == v = abs(sign(u - v))*2-1 # assuming sign(0) == 0 x?u:v = (u + v + x*(u - v))/2 = (u*(1 + x) + v*(1 - x))/2

### RYNO8 commented Jun 28, 2021

also:
min(max(a,b),c)=max(min(a,c),b)

min(min(a,b),b)=min(a,b)
max(a+c,b+c)=c+max(a,b)