Comment #8 on issue 3006 by [email protected]: Inaccurate sin/cos values
http://code.google.com/p/v8/issues/detail?id=3006
A few things to consider:
- The ECMA script specification clearly states that Math.sin/cos are
implementation-dependent approximations. There is no guarantees required
regarding precision.
- The glibc implementation of sin and cos itself is inconsistent depending
on whether it's the 32 bit version using FPU or the 64 bit version using
SSE. It's not accurate by your definition at all.
- In case of cos(2^120), is 0 more or less precise than
0.47796506772457525, when the actual result should be roughly -0.925879?
- Digital sampling inherently accepts rounding errors. I could argue that
always rounding down is less precise than rounding to the nearest. In that
case, rounding -7e-14 to 0 is more precise.
- The new implementation of Math.tan(Math.PI/2) returns Infinity, not
16331778728383844. Is that closer to the truth?
- I can assure you that my current approximation is a lot more accurate
than 2-3 terms of Taylor expansion.
- Math.PI is only a double precision approximation to begin with, namely
3.141592653589793. What comes after would be ...2384626..., an error of
~2.38e-16. That multiplied by 2200 is an error of ~4.7e-13, That seems a
lot worse than -7e-14.
- The approximation is not simply a table lookup of one value. It's a cubic
spline interpolation depending on two closest samples including
derivatives. The test case I wrote alongside the new approximation uses
tailor expansions with 25 terms as reference. Testing it against a set of
randomly and manually chosen values, the error rms of glibc is 8.696e-14,
the error rms of the new approximation is 8.723e-14. Yes, if you compare
the result bit patterns, they do differ, but the one is just a good guess
as the other. I gladly accept the negligible loss of precision for more
than twice the performance.
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