On 2013-09-12, at 1:17 AM, Dmitry Nadezhin <dmitry.nadez...@gmail.com> wrote: > The optimal constant for double conversion could be 768 , > the optimal constant for float conversion could be 142, > but I leave this optimization to JDK 9.
It would be helpful to mention in the proof/comment, that 768 refers to the decimal representation that has had leading zeroes between decimal point and mantissa trimmed. An exact case is the decimal representation of the sum of 2^-1022 + 2^-1075 This has a decimal point, then 307 zeroes, non-zeroes at 308 (2) and 1075 (5) (and usually non-zeroes between) for a span of 768 decimal digits between the leading and trailing zeroes. (There are other screw cases. Here, 2^-1022 is the minimum normalized-rep double, and 2^-1075 is 1/2 ulp for doubles with binary exponent -1022, which is represented as 1 in IEEE format.) Appending any trailing non-zero on this decimal representation rounds up; nothing trailing rounds down. If the final digit 5 is instead a 6, it always rounds up, if it is a 4 it always rounds down no matter what is appended. There are other screw cases, but they always involve 2^-1075 because that requires the maximum number of decimal digits for its representation. David
