[jhgb]

But am I correct in thinking that is at worst a 0.5 ulp error in this case? The lesser term in double-double can't be more than 0.5 ulp of the greater term and sensitivity of both sine and cosine to an error in the input will not be more than 1.

Also, in case of confusion, I was specifically commenting on the function over the [-pi/4, pi/4] domain in https://github.com/ifduyue/musl/blob/master/src/math/__cos.c , which the comment in https://news.ycombinator.com/item?id=30846546 was presumably about.

[lifthrasiir]

Yeah, sine and cosine are not as sensitive (but note that many libms target 1 or 1.5 ulp error for them, so a 0.5 ulp error might still be significant). For tangent however you definitely need more accurate range reduction.

[adgjlsfhk1]

Double rounding can still bite you. You are forced to incur up to half an ulp of error from your polynomial, so taking another half ulp in your reduction can lead to a total error of about 1 ulp.