Average Error: 6.1 → 5.1
Time: 2.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -4.55858928807585438 \cdot 10^{-16} \lor \neg \left(\frac{1}{x} \le 2.9766667581445495 \cdot 10^{122}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \le -4.55858928807585438 \cdot 10^{-16} \lor \neg \left(\frac{1}{x} \le 2.9766667581445495 \cdot 10^{122}\right):\\
\;\;\;\;\frac{1}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((1.0 / x) / (y * (1.0 + (z * z))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((((1.0 / x) <= -4.558589288075854e-16) || !((1.0 / x) <= 2.9766667581445495e+122))) {
		VAR = (1.0 / (y * ((1.0 + (z * z)) * x)));
	} else {
		VAR = (((1.0 / y) / x) / (1.0 + (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

Original6.1
Target5.5
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 x) < -4.558589288075854e-16 or 2.9766667581445495e+122 < (/ 1.0 x)

    1. Initial program 12.4

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

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

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

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

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

    5. Applied times-frac12.4

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

    6. Applied times-frac9.7

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

    7. Simplified9.7

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\]
    8. Using strategy rm

    9. Applied div-inv9.7

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

    10. Applied associate-/l*9.7

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

    11. Simplified9.7

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

    13. Applied frac-times9.6

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

    14. Simplified9.6

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

    if -4.558589288075854e-16 < (/ 1.0 x) < 2.9766667581445495e+122

    1. Initial program 2.4

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

    3. Applied associate-/r*2.4

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

    4. Simplified2.4

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -4.55858928807585438 \cdot 10^{-16} \lor \neg \left(\frac{1}{x} \le 2.9766667581445495 \cdot 10^{122}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))