Average Error: 6.6 → 5.7
Time: 6.3s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le -1.29234051502767109 \cdot 10^{-42} \lor \neg \left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le 7.1428719458543589 \cdot 10^{-221}\right):\\ \;\;\;\;\frac{\frac{1}{1}}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le -1.29234051502767109 \cdot 10^{-42} \lor \neg \left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le 7.1428719458543589 \cdot 10^{-221}\right):\\
\;\;\;\;\frac{\frac{1}{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 ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))))) <= -1.2923405150276711e-42) || !(((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))))) <= 7.142871945854359e-221))) {
		VAR = ((double) (((double) (1.0 / 1.0)) / ((double) (y * ((double) (((double) (1.0 + ((double) (z * z)))) * x))))));
	} else {
		VAR = ((double) (((double) (((double) (1.0 / y)) / x)) / ((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

Original6.6
Target5.9
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) < -1.2923405150276711e-42 or 7.142871945854359e-221 < (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied *-un-lft-identity0.3

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

    5. Applied times-frac0.3

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

    6. Applied times-frac1.0

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

    7. Simplified1.0

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

    9. Applied associate-*l/1.0

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

    10. Simplified1.0

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

    12. Applied *-un-lft-identity1.0

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

    13. Applied div-inv1.0

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

    14. Applied times-frac1.0

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

    15. Applied associate-/l*1.0

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

    16. Simplified1.0

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

    if -1.2923405150276711e-42 < (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))) < 7.142871945854359e-221

    1. Initial program 11.0

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

    3. Applied associate-/r*8.9

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

    4. Simplified8.9

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le -1.29234051502767109 \cdot 10^{-42} \lor \neg \left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \le 7.1428719458543589 \cdot 10^{-221}\right):\\ \;\;\;\;\frac{\frac{1}{1}}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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)))))