x = 0.5
for k in range(10): print k,x; x = log(5*sin(2*pi*x))

0 0.5
1 -35.0292711003
/usr/bin/ipython:2: RuntimeWarning: invalid value encountered in log

x = 0.5
for k in range(10): print k,x; x = arcsin(exp(x)/5)/(2*pi)

0 0.5
1 0.05348125972
2 0.033833959096
3 0.0331659027955
4 0.0331434275576
5 0.0331426717019
6 0.0331426462824
7 0.0331426454275
8 0.0331426453988
9 0.0331426453978