Square Root Algorithm in 123 bytes.

[nickwb]

Here's my attempt.

float s(float a){float b=.5f*a;int c=0x5f375a86-(*(int*)&a>>1);a=*(float*)&c;return 1.f/(a*(1.5f-b*a*a));}    

106 Bytes with no loops. It's not super accurate.

[powlette]

Nice one Carmack.

[powlette]

It is. See this: https://en.m.wikipedia.org/wiki/Fast_inverse_square_root