php, 56 bytes

Result

WA

Code [DL]

<?php for(;$c=fgets(STDIN);){echo sqrt(8*bindec($c)+1);}

stdin

00101110
10011001
00111011
01000010
10101011
01011111
10110010
10100110
00010101
11100111
00000110
00100100
10111111
00000011
11001100
00001000
10011111
01000100
00000000
01101001
11111101
00001100
10001000
00001010
11011011
01111000
10111110
11100111
00100000
11011100
00011001
00101101
00100001
01001110
11101111
00100100
00000100
11100100
00001111
00000001
11000010
00110111
00100111
11010011
00101011
00011100
00101010
01011011
11010010
10001111

stdout

19.2093727122993521.748563170932233727.58622844826737.74917217635436.455452267116134371739.102429592034540.4103947023538.062257748298535.67912554982323.345235059858129459.848857801796133941.86884283091731394316.03121954188141.96427051671514.1774468787581916.27882059612543.737855457258175.74456264653842.72001872658811339.4081209904762117.69180601295441.09744517606918.5741756210071518.357559750686274133.837848631377

stderr

strace