Specify correct rounding for sqrt

[hpkfft]

In accuracy.rst, correct rounding is required for add, sub, mul, and divide. I propose adding sqrt to this list. Note that sqrt is not listed later among the mathematical functions whose accuracy is not precisely defined.

Note that the IEEE Std 754-2019 for Floating-Point Arithmetic mandates correct rounding for squareRoot.

[asmeurer]

Is this something that's actually the case for existing libraries/devices? If so, there shouldn't be an issue with adding it.

By the way, is your proposal for this to apply to both real and complex inputs or just to real inputs?

[hpkfft]

Excellent point! My proposal is for this to apply just to real inputs. Given industry adoption of IEEE Std 754-2019, correct rounding is actually the case for existing libraries/devices. Sometimes there is an option for less accuracy (and presumably higher performance). For example, CUDA in single precision requires compiling with -prec-sqrt=true, and division requires -prec-div=true, for correct rounding. Since this document requires correctly rounded division, it equally well ought to require correctly rounded square root. [Double precision CUDA divide and square root are always correctly rounded. See the "Mathematical Functions" section of the "CUDA C++ Programming Guide".]

[hpkfft]

I believe the existing accuracy requirements for element-wise arithmetic operations are only intended to apply to real inputs. For example, the usual 6 flop implementation of complex multiplication

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

does not result in correctly rounded real and imaginary components. I do not believe this array standard means to prohibit this implementation, but this should be clarified.