Haskell sine and cosine functions not working

Question

Okay, this is a pretty weird problem. The built-in Haskell sine function (sin) does not seem to work. `sin 0` gives, correctly, `0`. `sin pi` gives, for whatever reason, `1.2246467991473532e-16` These are using the built in prelude functions. I simply start up ghci (the Haskell interpreter), and type in `sin pi` and get the wrong answer. Also, `cos (pi/2)` gives `6.123233995736766e-17` Any ideas why this might be? It looks like the build in functions are simply wrong.. which seems extremely unlikely seeing how mathematically-oriented the Haskell standard library is. edit: Heh, I just simply overlooked the e-16.. I guess that's what I get for coding late at night. Thanks anyhow everyone!

Accepted Answer

This is Matlab

```
>> sin(pi)
ans =
  1.2246e-016
```

And here's Python

```
>>> from math import sin, pi
>>> sin(pi)
1.2246467991473532e-16
```

You're running into the limits of floating point precision. I recommend having a read of What Every Computer Scientist Should Know About Floating Point Arithmetic.

The `e` at the end of these numbers indicates that they're in (a compact form of) scientific notation, and stands for "× 10^". For instance, in this notation, `2e3` corresponds to 2 × 103 = 2000. Here, you have a number that's multiplied by 10-16, which is tiny; written out in full, `1.2246467991473532e-16` = 0.00000000000000012246467991473532, so the amount of error is very small.

If you want accurate real-number computations in Haskell, you can use the `CReal` package as follows.

```
>>> import Data.Number.CReal
>>> sin (0.0  :: CReal)
0.0
>>> sin (pi   :: CReal)
0.0
>>> cos (pi/2 :: CReal)
0.0
```

This works because "under the hood" a `CReal` is a function `Int -> Integer`. Given a number of digits `d` to output, the function produces the `Integer` that, when divided by `10^d`, would give the real number correct to `d` decimal places.

Haskell sine and cosine functions not working

Solution

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