Computing the error bound of floating-point expression

Question

How should I compute the maximum absolute and relative error of the following IEEE-754 floating-point expression?

a.y + (x - a.x) * ((b.y - a.y) / (b.x - a.x))

Also, we assume, that

• the optimizer leaves the expression in the specified form
• the default rounding mode (round to nearest even)

Answer 1

IEEE 754 gives you the maximum relative error of each operation. You have three differences with a maximum relative error each. Multiplication and division adds the maximum relative error of each operand, plus the maximum relative error of the multiplication and division.

Then you have a final addition. If the right hand side is very close to the negative of the left hand side of the "+" then the relative error is unlimited. You'd need to analyse how close to -a.y the right hand side can be.

For absolute error, you convert the relative error of the right hand side into an absolute error.

Answer 2

Let the absolute errors on all variables be equal to $\delta$.

The absolute errors on the differences are $2\delta$ and the relative errors on the second term is

$$\frac{2\delta}{x-a_x}+\frac{2\delta}{b_y-a_y}+\frac{2\delta}{b_x-a_x}$$

So the final absolute error,

$$\delta\left(1+2\frac{(x-a_x)(b_y-a_y)}{b_x-a_x}\left(\frac1{x-a_x}+\frac1{b_y-a_y}+\frac1{b_x-a_x}\right)\right)$$

or

$$\delta\left(1+2\left(\frac{b_y-a_y}{b_x-a_x}+\frac{x-a_x}{b_x-a_x}+\frac{b_y-a_y}{b_x-a_x}\frac{x-a_x}{b_x-a_x}\right)\right)$$