numpy division with RuntimeWarning: invalid value encountered in double_scalars

Question

I wrote the following script:

import numpy

d = numpy.array([[1089, 1093]])
e = numpy.array([[1000, 4443]])
answer = numpy.exp(-3 * d)
answer1 = numpy.exp(-3 * e)
res = answer.sum()/answer1.sum()
print res

But I got this result and with the error occurred:

nan
C:\Users\Desktop\test.py:16: RuntimeWarning: invalid value encountered in double_scalars
  res = answer.sum()/answer1.sum()

It seems to be that the input element were too small that python turned them to be zeros, but indeed the division has its result.

How to solve this kind of problem?

Answer 1

You can't solve it. Simply answer1.sum()==0, and you can't perform a division by zero.

This happens because answer1 is the exponential of 2 very large, negative numbers, so that the result is rounded to zero.

nan is returned in this case because of the division by zero.

Now to solve your problem you could:

• go for a library for high-precision mathematics, like mpmath. But that's less fun.
• as an alternative to a bigger weapon, do some math manipulation, as detailed below.
• go for a tailored scipy/numpy function that does exactly what you want! Check out @Warren Weckesser answer.

Here I explain how to do some math manipulation that helps on this problem. We have that for the numerator:

exp(-x)+exp(-y) = exp(log(exp(-x)+exp(-y)))
                = exp(log(exp(-x)*[1+exp(-y+x)]))
                = exp(log(exp(-x) + log(1+exp(-y+x)))
                = exp(-x + log(1+exp(-y+x)))

where above x=3* 1089 and y=3* 1093. Now, the argument of this exponential is

-x + log(1+exp(-y+x)) = -x + 6.1441934777474324e-06

For the denominator you could proceed similarly but obtain that log(1+exp(-z+k)) is already rounded to 0, so that the argument of the exponential function at the denominator is simply rounded to -z=-3000. You then have that your result is

exp(-x + log(1+exp(-y+x)))/exp(-z) = exp(-x+z+log(1+exp(-y+x)) 
                                   = exp(-266.99999385580668)

which is already extremely close to the result that you would get if you were to keep only the 2 leading terms (i.e. the first number 1089 in the numerator and the first number 1000 at the denominator):

exp(3*(1089-1000))=exp(-267)

For the sake of it, let's see how close we are from the solution of Wolfram alpha (link):

Log[(exp[-3*1089]+exp[-3*1093])/([exp[-3*1000]+exp[-3*4443])] -> -266.999993855806522267194565420933791813296828742310997510523

The difference between this number and the exponent above is +1.7053025658242404e-13, so the approximation we made at the denominator was fine.

The final result is

'exp(-266.99999385580668) = 1.1050349147204485e-116

From wolfram alpha is (link)

1.105034914720621496.. × 10^-116 # Wolfram alpha.

and again, it is safe to use numpy here too.