Operations Specified By The Standards

Handling Signed 0

If $x,y$ are nonzero, and $x+y=0$ or $x-y=0$ exactly, then it is $+0$ for RN, RZ, RU modes, and $-0$ for RD.

However, if we have $x+x$ or $x-(-x)$, and $x=\pm0$, then the result has the same sign as $x$.

So be careful. If you have $x+0$, the result is not necessarily $x$. If $x=-0$, then the result is $+0$ in RN.

Square Root

If the input is greater than or equal to 0, the result always has a positive sign, with the exception of $\sqrt{-0}=-0$ Remainders Let $x$ be a finite floating point number. Let $y$ be a nonzero, finite floating point number. Then the remainder $r=x\mathrm{REM}y$ is defined as:

1. $r=x-yn$ where $n$ is the integer closest to $x/y$.
2. If $x/y$ half way between two integers, let $n$ be the even integer.
3. If $r=0$, its sign is that of $x$.

Note: Remainders are always exactly representable under these conditions.

Also note that nowhere does it say $x,y$ need to be integers!

Finally, $x\mathrm{REM}\infty=x$. Preferred Exponent How do we decide what the exponent should be for the result of an operation? This is not obvious for decimal formats. The rules are:

1. If the result is inexact, use the smallest exponent.
2. If the result is exact, then if the cohort includes a preferred exponent (defined below), use that one. Otherwise pick the one with the closest exponent to the preferred one.

Below are the preferred exponents:

• $x\pm y$: The smaller of the two exponents
• $xy$: Add the two exponents
• $x/y$: Subtract the two exponents
• $\FMA(x,y,z)$: $\min(Q(x)+Q(y),Q(z))$
• $\sqrt{x}$: $\lfloor Q(x)/2\rfloor$ scaleB and logB

$\mathrm{scaleB}(x,n)$ is defined as $x\beta^{n}$, correctly rounded, where $x$ is any floating point number and $n$ is an integer.

When $x$ is finite and nonzero, $\mathrm{logB}(x)$ is $\lfloor\log_{\beta}|x|\rfloor$. We also have $\mathrm{logB}(\NaN)=\NaN$, $\mathrm{logB}(\pm0)=-\infty$, $\mathrm{logB}(\pm\infty)=\infty$