Average Error: 12.4 → 2.3
Time: 2.1s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3343627764232323 \cdot 10^{30}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;x \le 1.1988184646526967 \cdot 10^{-249}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3343627764232323 \cdot 10^{30}:\\
\;\;\;\;\frac{x}{\frac{z}{y + z}}\\

\mathbf{elif}\;x \le 1.1988184646526967 \cdot 10^{-249}:\\
\;\;\;\;\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 VAR;
	if ((x <= -1.3343627764232323e+30)) {
		VAR = (x / (z / (y + z)));
	} else {
		double VAR_1;
		if ((x <= 1.1988184646526967e-249)) {
			VAR_1 = (((x * y) / z) + x);
		} else {
			VAR_1 = (x * ((y + z) / z));
		}
		VAR = VAR_1;
	}
	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
Target3.0
Herbie2.3
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3343627764232323e+30

    1. Initial program 23.9

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

    3. Applied associate-/l*0.1

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

    if -1.3343627764232323e+30 < x < 1.1988184646526967e-249

    1. Initial program 5.6

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

    2. Taylor expanded around 0 3.2

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

    if 1.1988184646526967e-249 < x

    1. Initial program 13.3

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

    3. Applied *-un-lft-identity13.3

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

    4. Applied times-frac2.4

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

    5. Simplified2.4

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3343627764232323 \cdot 10^{30}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;x \le 1.1988184646526967 \cdot 10^{-249}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))