I am performing a probability computation. I have many very very small numbers, all of which I want to subtract from 1, and do so accurately. I can accurately compute the logarithm of these small numbers. My strategy so far has been like so (using numpy):
Given an array of the log of the small numbers x, compute:
y = numpy.logaddexp.reduce(x)
Now I want to compute something like 1-exp(y) or even better log(1-exp(y)), but I'm not sure how to do so without losing all my precision.
In fact, even the logaddexp function is running into precision problems. Values in the vector x can range from -2 through -800, or even more negative. The vector y from above would basically have a whole section of numbers around 1e-16, which is the eps of the data type. So, for example, the accurately computed data could look like this:
In [358]: x
Out[358]:
[-5.2194676211172837,
-3.9050377656308362,
-3.1619783292449615,
-2.71289594096134,
-2.4488395891021639,
-2.3129210706827568,
-2.2709987626652346,
-2.3007776073511259,
-2.3868404149802434,
-2.5180718876609163,
-2.68619816583087,
-2.8849022632856958,
-3.1092603032627686,
-3.3553673369747834,
-3.6200806272462351,
-3.9008385919463073,
-4.1955300857178379,
-4.5023981074719899,
-4.8199676154248081,
-5.1469905756384904,
-5.4824035553480428,
-5.8252945959126876,
-6.174877049340779,
-6.5304687083067563,
-6.8914750074202473,
-7.25737538919104,
-7.6277121540338797,
-8.0020812775389558,
-8.3801247986220773,
-8.7615244716292437,
-9.1459964426584435,
-9.5332867613176404,
-9.9231675781398394,
-10.315433907978701,
-10.709900863130784,
-11.106401278287066,
-11.50478366390567,
-11.904910436107656,
-12.30665638039909,
-12.709907313918777,
-13.114558916892051,
-13.52051570882999,
-13.927690148982549,
-14.336001843810081,
-14.745376846921289,
-15.15