Surreal Numbers

Question

Surreal Numbers

Surreal numbers are one way of describing numbers using sets. In this challenge you will determine the value of a surreal number.

Intro

A surreal number consists of two sets: a left and right. The value of the surreal number must be greater than all numbers in the left set and less than all numbers in the right set. We define 0 as having nothing in each set, which we write as {|}. Each surreal number also has a birth date. 0 is born on day 0. On each successive day, new surreal numbers are formed by taking the numbers formed on previous days and placing them in the left/right sets of a new number. For example, {0|} = 1 is born on day 1, and {|0} = -1 is also born on day 1.

To determine the value of a surreal number, we find the number "between" the left and right sets that was born earliest. As a general rule of thumb, numbers with lower powers of 2 in their denominator are born earlier. For example, the number {0|1} (which is born on day 2 by our rules) is equal to 1/2, since 1/2 has the lowest power of 2 in its denominator between 0 and 1. In addition, if the left set is empty, we take the largest possible value, and vice versa if the right set is empty. For example, {3|} = 4 and {|6} = 5.

Note that 4 and 5 are just symbols that represent the surreal number, which just happen to align with our rational numbers if we define operations in a certain way.

Some more examples:

{0, 1|} = {1|} = {-1, 1|} = 2
{0|3/4} = 1/2
{0|1/2} = 1/4
{0, 1/32, 1/16, 1/2 |} = 1

Input

A list/array of two lists/arrays, which represent the left and right set of a surreal number. The two lists/arrays will be sorted in ascending order and contain either dyadic rationals (rationals with denominator of 2^n) or other surreal numbers.

Output

A dyadic rational in either decimal or fraction form representing the value of the surreal number.

Examples

Input -> Output
[[], []] -> 0
[[1], []] -> 2
[[], [-2]] -> -3
[[0], [4]] -> 1
[[0.25, 0.5], [1, 3, 4]] -> 0.75
[[1, 1.5], [2]] -> 1.75
[[1, [[1], [2]]], [2]] -> 1.75

This is code-golf, so shortest code wins.

Answer 1

Python, 371 362 261 251 248 241 236 234 231 bytes

-122 bytes thanks to @ASCII-only

def f(s):
 if s<[]:return s
 a=map(f,s[1]+[1e999,-1e999]+s[0]);m,n,u=a[-1],a[0],1
 while n-m<=1:m*=2;n*=2;u/=2.
 x,y=abs(m)-1,abs(n)-1;s=m>0>n or m<0<n;r=[[m-x*s,m+x*s][m<0]+1,[n-y*s,n+y*s][n<0]-1][x>y];return u*int([0,r][`r`<"m"])

Try it online!