Average Error: 12.4 → 3.3
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.368950694637007 \cdot 10^{-262} \lor \neg \left(z \le 1.47174836325826194 \cdot 10^{-149}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y + z}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.368950694637007 \cdot 10^{-262} \lor \neg \left(z \le 1.47174836325826194 \cdot 10^{-149}\right):\\
\;\;\;\;x \cdot \frac{y + z}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y + z}}\\

\end{array}
double code(double x, double y, double z) {
	return ((x * (y + z)) / z);
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -3.368950694637007e-262) || !(z <= 1.471748363258262e-149))) {
		VAR = (x * ((y + z) / z));
	} else {
		VAR = ((x / z) / (1.0 / (y + 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
Target3.1
Herbie3.3
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.368950694637007e-262 or 1.471748363258262e-149 < z

    1. Initial program 12.7

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

    3. Applied *-un-lft-identity12.7

      \[\leadsto \frac{x \cdot \left(y + z\right)}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac2.0

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

    5. Simplified2.0

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

    if -3.368950694637007e-262 < z < 1.471748363258262e-149

    1. Initial program 10.5

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

    3. Applied associate-/l*12.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]
    4. Using strategy rm

    5. Applied div-inv12.8

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{y + z}}}\]

    6. Applied associate-/r*13.3

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.368950694637007 \cdot 10^{-262} \lor \neg \left(z \le 1.47174836325826194 \cdot 10^{-149}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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))