# Implied precision of floating-point functions

While reviewing another programmer's implementation of a function to calculate the normal distribution CDF, I made a suggestion to either replace the entire implementation with Python's built-in functions or use SciPy, a common scientific library.

Another programmer pointed out that neither `math.erfc()` nor `scipy.stats.norm.cdf()` provides any precision guarantees in their documentation. Therefore, I should be more cautious about replacing an approximation algorithm (which was taken from a respected source, and which had documented error bounds).

To be honest, the thought to doubt the accuracy and precision of a built-in or library function had never crossed my mind. After all, I've been calling functions like `sin()` and `sqrt()` for years without much thought — why should `math.erf()` or `scipy.stats.norm.cdf()` be any different?

But now, I'm concerned. My questions are:

1. In general, if the documentation makes no special mention, is it implied that these kinds of functions are completely accurate to the last decimal place, within the precision offered by IEEE double-precision floating-point?
2. Is that true for Python's `math.erf()` or SciPy's `scipy.stats.norm.cdf()` in particular? How can you tell?
3. This man page for `sin()` says…

These functions may lose accuracy when their argument is near a multiple of pi or is far from 0.0.

Why should such caveats exist, when the sine function is periodic and symmetrical? There seems to be a burden placed on the caller to canonicalize the input to obtain optimal accuracy.

On the other hand, Mozilla's documentation for `Math.sin()` says nothing about accuracy or precision. Does that mean that it is fully accurate, or is it "common knowledge" that `Math.sin()` would only be accurate in certain circumstances in JavaScript, like everywhere else?

• FYI regarding question 1: Usually precision guarantees are given in terms of ULP (units in the last place) which refers to the binary digits of the float. – user7043 Aug 8 '14 at 19:23

If the documentation makes no special mention, is it implied that these kinds of functions are completely accurate to the last decimal place, within the precision offered by IEEE double-precision floating-point?

I wouldn't make that assumption.

Where I work we deal with telemetry data, and it's common knowledge that two different math libraries can produce two different results, even if they both conform to the IEEE floating point standards. This has implications when you're trying to repeat a calculation and compare two results for equality.

There seems to be a burden placed on the caller to canonicalize the input to obtain optimal accuracy.

That's a fair assessment. What makes it acceptable is that the documentation states that, so that there are no surprises.

On the other hand, Mozilla's documentation ...

I suggest that you try a few calculations in each library or programming language (especially near the boundary areas like `sin()` near a multiple of `pi`), and compare them. This should give you a fair idea of what sort of behavior you can expect from each one.

• Test is a good idea. There are countless times where the documentation says one thing, yet the functions behave another way. And OP want to rely on implicit assumptions that are not even documented. – Siyuan Ren Aug 9 '14 at 3:42