Xcell Journal Issue 78: Charge to Market with Xilinx 7 Series ...

XPLANATION: FPGA 101
Parameter Binary format (base, b = 2) Decimal format (base, b = 10)
need to take care of certain things before finalizing the
design. We need to examine if numbers from the data set
can be very close in floating-point representation, very big
or very small. Let us consider the roots of a quadratic equation
as an example. The two roots, α and β are given by the
following two formulas:
α=(-b+√(b^2-4ac))/2a ; β= (-b-√(b^2-4ac))/2a
Clearly b^2 and 4ac will be subject to rounding errors
when they are floating-point numbers. If b^2 >> 4ac,
then √(b^2 ) = |b|. Additionally, if b > 0, β will give some
result but α will give zero as result because the terms in its
numerator get canceled. On the other hand, if b < 0, then β
will be zero while α will give some result. This is erroneous
design, since roots that become zero unintentionally
because of floating-point calculations, when used later in
some part of the design, can easily cause a faulty output.
One way to prevent such erroneous cancellation is to
multiply the numerator and denominator of α and β by
-b-√(b^2-4ac). Doing so will give us:
α’ = 2c/(-b-√(b^2-4ac)) ; β’= 2c/(-b+√(b^2-4ac))
Now, when b>0, we should use α’ and β to get the values of
the two roots and when b