# Multiply with restricted operations

### 22 operations

itx = 1/(1+a+b)     #4
nx = -1/(itx+itx)   #4
c = -( 1/(itx + itx + 1/(1+nx)) + 1/(1/(a+nx) + 1/(b+nx)) ) #14


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The ops are 10 additions, 7 inverses, 2 negations, and 3 assignments.

So, how did I get this? I started with the promising-looking template of the sum of two double-decker fractions, a motif that had appeared in many previous attempts.

c = 1/(1/x + 1/y) + 1/(1/z + 1/w)

When we restrict the sum to x+y+z+w=0, a beautiful cancellations occur, giving:

c = (x+z)*(y+z)/(x+y),

which contains a product. (It's often easier to get t*u/v rather than t*u because the first has degree 1.)

There's a more symmetric way to think about this expression. With the restriction x+y+z+w=0, their values are specified by three parameters p,q,r of their pairwise sums.

 p = x+y
-p = z+w
q = x+z
-q = y+w
r = x+w
-r = y+z


and we have c=-q*r/p. The sum p is distinguished as being in the denominator by corresponding to the pairs (x,y) and (z,w) of variables that are in the same fraction.

This is a nice expression for c in p,q,r, but the double-decker fraction is in x,y,z,w so we must express the former in terms of the latter:

x = ( p + q + r)/2
y = ( p - q - r)/2
z = (-p + q - r)/2
w = (-p - q + r)/2


Now, we want to choose p,q,r so that c=-q*r/p equals a*b. One choice is:

p = -4
q = 2*a
r = 2*b


Then, the doubled values for q and r are conveniently halved in:

x = -2 + a + b
y = -2 - a - b
z =  2 + a - b
w =  2 - a + b


Saving 2 as a variable t and plugging these into the equation for c gives a 24-op solution.

#24 ops
t = 1+1   #2
c = 1/(1/(-t+a+b) + 1/-(t+a+b))  +  1/(1/(-b+t+a) + 1/(-a+b+t)) #1, 10, 1, 10


There's 12 additions, 6 inverses, 4 negations, and 2 assignments.

A lot of ops are spent expressing x,y,z,w in terms of 1,a,b. To save ops, instead express x in p,q,r (and thus a,b,1) and then write y,z,w in terms of x.

y = -x + p
z = -x + q
w = -x + r


Choosing

p = 1
q = a
r = b


and expressing c with a negation as c=-q*r/p, we get

x = (1+a+b)/2
y = -x + 1
z = -x + a
w = -x + b


Unfortunately, halving in x is costly. It needs to be done by inverting, adding the result to itself, and inverting again. We also negate to produce nx for -x, since that's what y,z,w use. This gives us the 23-op solution:

#23 ops
itx = 1/(1+a+b)     #4
nx = -1/(itx+itx)   #4
c = -( 1/(1/(-nx) + 1/(1+nx))  +  1/(1/(a+nx) + 1/(b+nx)) ) #15


itx is 1/(2*x) and nx is -x. A final optimization of expressing 1/x as itx+itx instead of the templated1/(-nx) cuts a character and brings the solution down to 22 ops.

## 23 operations

z = 1/(1/(1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1))-(1/a+1/b))
res = z+z


proof by explosion:

z = 1/(1/(1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1))-(1/a+1/b))
1/(a+1)+1/(b+1)                            == (a+b+2) / (ab+a+b+1)
1/(1/(a+1)+1/(b+1))                           == (ab+a+b+1) / (a+b+2)
1/(1/(a+1)+1/(b+1))-1                         == (ab - 1) / (a+b+2)
1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1)             == ab / (a+b+2)
1/(1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1))            == (a+b+2) / ab
1/a+1/b   == (a+b) / ab
1/(1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1))-(1/a+1/b)  == 2 / ab
1/(1/(1/(1/(a+1)+1/(b+1))-1+1/(a+b+1+1))-(1/a+1/b)) == ab / 2

z = ab / 2 and therefore z+z = ab


I abused wolfram alpha to get this beautiful image (wolfram alpha tried to get me to subscribe to pro to save it, but then ctrl-c ctrl-v ;-)):

score (with added + on subtraction):

z = ////++/++-+/++++-/+/
res = +


## 29 operations

Does not work for the set $$\\{(a,b) \in \mathbb{R}^2 : a\pm b=0 \text{ or } a\pm b = -1\}\$$. That's probably measure zero?

sum = a+b
nb = -b
diff = a+nb
rfc = 1/(1/(1/sum + -1/(sum+1)) + -1/(1/diff + -1/(diff+1)) + nb + nb)  # rfc = 1/4c
c = 1/(rfc + rfc + rfc + rfc)

# sum  is  2: =+
# nb   is  2: =-
# diff is  2: =+
# rfc  is 18: =///+-/++-//+-/+++
# c    is  5: =/+++
# total = 29 operations


The structure of rfc (Reciprocal-Four-C) is more evident if we define a macro:

s(x) = 1/(1/x + -1/(x+1))              # //+-/+ (no = in count, macros don't exist)
rfc = 1/(s(sum) + - s(diff) + nb + nb) # =/s+-s++ (6+2*s = 18)


Let's do the math:

• s(x), mathematically, is 1/(1/x - 1/(x+1)) which is after a bit of algebra is x*(x+1) or x*x + x.
• When you sub everything into rfc, it's really 1/((a+b)*(a+b) + a + b - (a-b)*(a-b) - a + b + (-b) + (-b)) which is just 1/((a+b)^2 - (a-b)^2).
• After difference of squares, or just plain expansion, you get that rfc is 1/(4*a*b).
• Finally, c is the reciprocal of 4 times rfc, so 1/(4/(4*a*b)) becomes a*b.