On Wed, 14 Mar 2012, Vincent Lefevre wrote: > For double-double (IBM long double), I don't think the notion of > correct rounding makes much sense anyway. Actually the double-double > arithmetic is mainly useful for the basic operations in order to be > able to implement elementary functions accurately (first step in > Ziv's strategy, possibly a second step as well). IMHO, on such a > platform, if expl() (for instance) just calls exp(), this is OK.
expl just calling exp - losing 53 bits of precision - seems rather extreme. But I'd think it would be fine to say: when asked to compute f(x), take x' within 10ulp of x, and return a number within 10ulp of f(x'), where ulp is interpreted as if the mantissa were a fixed 106 bits (fewer bits for subnormals, of course). (And as a consequence, accurate range reduction for large arguments would be considered not to matter for IBM long double; sin and cos could return any value in the range [-1, 1] for sufficiently large arguments.) > > (b) Where functions do make attempts at being correctly rounded > > (especially the IBM Accurate Mathematical Library functions), they tend to > > be sufficiently slow that the slowness attracts bug reports. Again, this > > would likely be addressed by new implementations that use careful error > > bounds and information about worst cases to reduce the cost of being > > correctly rounding. > > I'm not sure that the complaints are about worst cases. More probably > software implementation vs hardware implementation in the average > case. But a new software implementation (better in average) could > help. Various bugs do complain about particular cases being slow (as well as about such things as sinf being slower than sin - there, if you automatically generate functions based not just on the type for the function being generated but also on what wider types are available and efficient in hardware, you could generate a version of sinf that uses double or long double computations internally to speed things up). -- Joseph S. Myers jos...@codesourcery.com
