I have an exercise based on relative-absolute errors and Bolzano-false point method. Exercise says that a straight line that passes through 2 points $(x_0,y_0)$ and $(x_1,y_1)$. So whenever $y=0$ ( cross point with x-axis) 1
$$ x=\frac{x_0×y_1−x_1×y_0}{y_1−y_0} ~~\text{ and }~~ x=x_0−(x_1−x_0)×{y_0\over y_1−y_0} $$ So, the sub-question asks to find which type of the above is more accurate using $(x_0,y_0)=(1.46,1.69)$ and $(x_1,y_1)=(2.58,3.37)$. I find that using the first type $x=0.33$ (using 3-digit significant round off) and $x=0.34$ with the second type. But I can't explain which type is more accurate. ( I know that I have to use False point method to prove it but I don't know how :P)
You have to use all the properties of the false position method. The most important assumption is that the values at $x_0$ and $x_1$ always have opposite sign. The important technical condition you have to ensure is that the computed $x$ is always inside the interval between $x_0$ and $x_1$. Small deviations of $x$ are not important as long as it is really inside the interval.
The opposite sign makes it so that in exact arithmetic $x$ is a convex combination of $x_0$ and $x_1$ with weights $|f(x_1)|$ and $|f(x_0)|$.
There is some slight probability in the first formula that for $|f(x_1)|\ll |f(x_0)|$ or vice versa that the floating point errors of multiplication and division do not exactly cancel out and $x$ lands outside the interval. In the second formula, the quotient $q=f(x_0)/(f(x_0)-f(x_1))$ is always in $[0,1]$ (remember, opposite sign), so that $x_0+q(x_1-x_0)$ is inside the segment.
In general, you will not observe such behavior in the regula falsi method due to its slow convergence. The numerical instability of the first formula is much more important in the secant method and related ones like Dekker's method, as there it is much more likely that the function values have the same sign and catastrophic cancellation occurs in the differences.