Average Error: 12.4 → 2.9
Time: 2.7s
Precision: binary64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.78412653436190541 \cdot 10^{-254} \lor \neg \left(z \le 2.32036261093918049 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.78412653436190541 \cdot 10^{-254} \lor \neg \left(z \le 2.32036261093918049 \cdot 10^{-59}\right):\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}\\

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) (y + z)))) / z));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -6.7841265343619054e-254) || !(z <= 2.3203626109391805e-59))) {
		VAR = ((double) (x / ((double) (z / ((double) (y + z))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (y + z)))) * ((double) (1.0 / z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.4
Target2.9
Herbie2.9
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.78412653436190541e-254 or 2.32036261093918049e-59 < z

    1. Initial program 13.3

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*1.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]

    if -6.78412653436190541e-254 < z < 2.32036261093918049e-59

    1. Initial program 8.8

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied div-inv9.0

      \[\leadsto \color{blue}{\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.78412653436190541 \cdot 10^{-254} \lor \neg \left(z \le 2.32036261093918049 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y + z\right)\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))