Average Error: 6.4 → 5.3
Time: 6.5s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -1.2219681213183615 \cdot 10^{38} \lor \neg \left(\frac{1}{x} \le 17176.799346420077\right):\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \le -1.2219681213183615 \cdot 10^{38} \lor \neg \left(\frac{1}{x} \le 17176.799346420077\right):\\
\;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((1.0 / x) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((((1.0 / x) <= -1.2219681213183615e+38) || !((1.0 / x) <= 17176.799346420077))) {
		VAR = ((double) ((1.0 / y) * ((1.0 / x) / ((double) (1.0 + ((double) (z * z)))))));
	} else {
		VAR = (((1.0 / x) / y) / ((double) (1.0 + ((double) (z * 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

Original6.4
Target5.7
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 x) < -1.2219681213183615e38 or 17176.799346420077 < (/ 1.0 x)

    1. Initial program 12.8

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity12.8

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

    4. Applied *-un-lft-identity12.8

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

    5. Applied times-frac12.8

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

    6. Applied times-frac10.2

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

    7. Simplified10.2

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

    if -1.2219681213183615e38 < (/ 1.0 x) < 17176.799346420077

    1. Initial program 1.8

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied associate-/r*1.7

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -1.2219681213183615 \cdot 10^{38} \lor \neg \left(\frac{1}{x} \le 17176.799346420077\right):\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))