Number of distinct non-empty subsequences of binary expansion
Python 3, 95 bytes  83 bytes

[-12 bytes thanks to Mr.XCoder :)]



def f(x):
 v=[2,1];c=1
 for i in bin(x)[3:]:k=int(i);c+=v[k];v[1-k]+=v[k]
 return c


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A note on the algorithm.
The algorithm computes the increment in unique subsequences given by the bit at a given position t.
The increment for the first bit is always 1. The algorithm then runs over the sequence of bits s(t) and adds the increment v[s(t)]. At each step, the increment for the complement of s(t), v[1 - s(t)] is updated to v[1]+v[0]. 
The final number is the sum of all increments. 

It should run in O(log2(n)), where n is the input number.