Re:C library for multiple-precision floating-point arithmeti...
by adrian_b on Friday June 03, 2022 @11:49AM
(#31609942)
Correct rounding means that if the function result has X bits, all those X bits should be the same as when computing the result with infinite precision and then rounding it using the specified rounding mode.
For the ordinary arithmetic operations, including for square root and for reciprocal square root, there are algorithms to do correct rounding without excessive computation. Because of that, the standard for floating-point arithmetic specifies that a conforming implementation must use correct rounding for those operations.
On the other hand, for transcedental functions, like exponential, logarithm, sine and so on, there is no general method to guarantee that the result will be correctly rounded without having to compute an unlimited number of result bits before being able to round correctly.
Because of the difficulty, many mathematical libraries and also all hardware implementations of transcedental functions round correctly many of the possible results, but not all of them.
Nevertheless, for each usual transcedental functions and for the numbers of bits used frequently, i.e. single-, double- and quadruple-precision, there are known algorithms to round correctly, which have usually been found by exhaustive searches for corner cases, when more result bits must be computed than for most other results.
For the ordinary arithmetic operations, including for square root and for reciprocal square root, there are algorithms to do correct rounding without excessive computation. Because of that, the standard for floating-point arithmetic specifies that a conforming implementation must use correct rounding for those operations.
On the other hand, for transcedental functions, like exponential, logarithm, sine and so on, there is no general method to guarantee that the result will be correctly rounded without having to compute an unlimited number of result bits before being able to round correctly.
Because of the difficulty, many mathematical libraries and also all hardware implementations of transcedental functions round correctly many of the possible results, but not all of them.
Nevertheless, for each usual transcedental functions and for the numbers of bits used frequently, i.e. single-, double- and quadruple-precision, there are known algorithms to round correctly, which have usually been found by exhaustive searches for corner cases, when more result bits must be computed than for most other results.