Average Error: 12.4 → 3.1
Time: 2.5s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 1.6319585875966188 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;z \le 7.99744230430140596 \cdot 10^{-211}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le 1.6319585875966188 \cdot 10^{-267}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\mathbf{elif}\;z \le 7.99744230430140596 \cdot 10^{-211}:\\
\;\;\;\;\frac{x \cdot y}{z} + x\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y + z}{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 temp;
	if ((z <= 1.6319585875966188e-267)) {
		temp = (x / (z / (y + z)));
	} else {
		double temp_1;
		if ((z <= 7.997442304301406e-211)) {
			temp_1 = (((x * y) / z) + x);
		} else {
			temp_1 = (x * ((y + z) / z));
		}
		temp = temp_1;
	}
	return temp;
}

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
Target3.2
Herbie3.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 1.6319585875966188e-267

    1. Initial program 12.3

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

    3. Applied associate-/l*3.5

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

    if 1.6319585875966188e-267 < z < 7.997442304301406e-211

    1. Initial program 11.5

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Taylor expanded around 0 7.8

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

    if 7.997442304301406e-211 < z

    1. Initial program 12.6

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

    3. Applied *-un-lft-identity12.6

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

    4. Applied times-frac2.2

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

    5. Simplified2.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1.6319585875966188 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;z \le 7.99744230430140596 \cdot 10^{-211}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))