Demonstrates an issue with the numerical stability of the gradient of arccos.

arccos_demo.py

import theano.tensor as tensor
from theano import function
import numpy as np
value = np.array([ 0.992827313923019039165751564724,
                   0.118152792712208978831434080803,
                  -0.008531793400639134036800292904,
                   0.016157066691618496290239193058])                           
aa = tensor.vector('aa')
bb = tensor.vector('bb')
ee = tensor.scalar()
cc = tensor.arccos(tensor.clip((aa * bb).sum(), -1, 1 - ee))
dd = tensor.grad(cc, bb)
g = function([aa, bb, ee], dd)
print g(value, value, 1e-7)
print g(value, value, 1e-8)
print g(value, value, 1e-9)
print g(value, value, 0)

output.txt

Output of the above Python code as of Theano master 11ee6cee3b:

wardefar@grieg:~$ python test.py 
[-0. -0.  0. -0.]
[-0. -0.  0. -0.]
[-0. -0.  0. -0.]
[-66627513.18554519  -7929099.69734382    572558.96299959
  -1084282.39124676]