Comment #6 on issue 3006 by [email protected]: Inaccurate sin/cos values
http://code.google.com/p/v8/issues/detail?id=3006
Responding to #3. You are making unwarranted assumptions on the meaning of
1000000 as a double float. 1000000 can be represented EXACTLY as a
double-precision IEEE754 floating point value. It is exactly 1000000 and no
other. cos(1000000) has exactly one value, roughly,
0.93675212753314478693853253507.... The same holds for 2^120. It has an
exact representation in double-precision and cos(2^120) has exactly one
value.
System libraries should strive to be as accurate as possible. If a user
wants to trade-off accuracy for speed, then it's quite easy. Doing it
accurately is hard, so the system implementation should be as accurate as
possible.
Consider what Kahan (primary architect of the IEEE754 standard) has to say:
http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf, section 13, page 25:
13 Prevalent Misconceptions about Floating-Point Arithmetic :
1• Floating–point numbers are all at least slightly uncertain.
And given that your current implementation has a large table of sin/cos
values, you would have more accurate results using sin(x) = sin(y+e) =
sin(y)*cos(e)+cos(y)*sin(e), where y is a value from your table. cos(e) and
sin(e) could be evaluated very accurately using 2-3 terms of the Taylor
expansion. This might be competitive in speed.
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