Theorem for FMA Let \(x,y,z\) be nonnegative floating point numbers. Assuming underflow does not occur, then \(xy+z\le...
The error of an FMA calculation is not always a floating point number. However, we can use two floating point numbers to exactly...
Suppose you need to multiply by a constant that is not exactly representable. Think \(\pi\) and the like. We’d like to multiply and...
The Basic Methods One way is to use Newton’s iteration on \(f(x)=x^{2}-a\). This method for calculating square root goes back thousands...
This section deals with floating point \(a,b\) values, not necessarily between 1 and 2. Assume they are non-negative, though. For this,...
We need to calculate \(o(a/b)\) where \(a,b\) are binary floating point numbers, and \(o\) is RN, RD, RU or RZ. We have a useful proof:...
Assume \(\beta=2\) for this section. Some of it may not work for decimal. We want to approximate \(b/a\). Assume \(1\le a,b<2\). In...
For this article, define a representable pair for a floating point number \(x\) to be any pair \((M,e)\) such that...
The 2MultFMA Algorithm This has been covered elsewhere. It works well when you use FMA. If No FMA Is Available If there is no FMA...
Let \(o\) be the rounding function, and \(a,b,c\) are floating point numbers. Then \(\mathrm{FMA}(a,b,c)\) is \(o(ab+c)\). if...