php, 64 bytes

Result

WA

Code [DL]

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

stdin

01111100
00110011
01000110
01001110
00011100
01011011
01100101
00101011
01100011
10111110
11101101
10100111
01101001
11010001
11100001
00110000
00000110
00101101
00100100
00000000
01000010
11110000
11011111
01011101
00010101
01111000
11001000
01000010
00000011
10101011
01101011
00001000
10001000
11111010
10001011
11100111
11111101
00110111
01101010
11111001
00001010
10011001
00000001
01001011
00110111
10100001
11010010
01000000
00001111
00010110

stdout

0.511902513177460.223748416156680.685438564654020000.442925306655780.574175621006710.1602556806574500.554563480765130.5650105975644500.902322672435120.438190347845890.62141687034858000000.829214001622250.249260348555220.29468812791236000.012498047485110000.274562336608890.06225774829854900.732538492690080.361654635224550000.137604568666930.643028571099430000.5153013442625300.9026461420324800.6495033058122500.30413469565007

stderr

strace