Re: [Haskell-cafe] Sinus in Haskell

Am Freitag, 9. November 2007 21:02 schrieb Hans van Thiel:

On Fri, 2007-11-09 at 14:30 -0500, Brent Yorgey wrote:

...

On Nov 9, 2007 2:08 PM, Hans van Thiel wrote: Hello All, Can anybody explain the results for 1.0, 2.0 and 3.0 times pi below? GHCi yields the same results. I did search the Haskell report and my text books, but to no avail. Thanks in advance, Hans van Thiel

Hugs> sin (0.0 * pi) 0.0 Hugs> sin (0.5 * pi) 1.0 Hugs> sin (1.0 * pi) 1.22460635382238e-16 Hugs> sin (1.5 * pi) -1.0 Hugs> sin (2.0 * pi) -2.44921270764475e-16 Hugs> sin ( 2.5 * pi) 1.0 Hugs> sin (3.0 * pi) 3.67381906146713e-16 Hugs>

More generally, this is due to the fact that floating-point numbers can only have finite precision, so a little bit of rounding error is inevitable when dealing with irrational numbers like pi. This problem is in no way specific to Haskell.

-Brent

All right, I'd have guessed that myself, if it hadn't been for the exact computation results for 0, 0.5, 1.5 and 2.5 times pi. So the rounding errors are only manifest for 1.0, 2.0 and 3.0 times pi. But look at the difference between sin (1.0 * pi) and sin (3.0 * pi). That's not a rounding error, but a factor 3 difference.. and sin (as well as cos) are modulo (2 * pi), right?

Regards, Hans

The exact results for 0, 0.5*pi and 2.5*pi aren't surprising. Leaving out the more than obvious case 0, we have two cases with sin x == 1. Now what's the smallest positive Double epsilon such that show (1+epsilon) /= show 1.0? I think, on my box it's 2^^(-52), which is roughly 2.22e-16, but the result calculated is closer to 1 than that, so the result is actually represented as 1.0, which by a lucky coincidence is the exact value. Regarding sin (1.0*pi) and sin (3.0*pi), I don't know how the sine is implemented,that they return nonzero values is expected, that actually sin (3.0*pi) == 3.0*sin pi does surprise me, too. Can anybody knowing more about the implementation of sine enlighten me? Cheers, Daniel