Average Error: 6.2 → 5.0
Time: 6.1s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.0165662251967751 \cdot 10^{30} \lor \neg \left(y \le 7943.81812641865054\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{y}}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.0165662251967751 \cdot 10^{30} \lor \neg \left(y \le 7943.81812641865054\right):\\
\;\;\;\;\frac{\frac{\frac{\frac{1}{y}}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}\\

\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 (((y <= -4.016566225196775e+30) || !(y <= 7943.8181264186505))) {
		VAR = ((double) (((double) (((double) (((double) (1.0 / y)) / x)) / ((double) sqrt(((double) (1.0 + ((double) (z * z)))))))) / ((double) sqrt(((double) (1.0 + ((double) (z * z))))))));
	} else {
		VAR = ((double) (1.0 / ((double) (((double) (y * ((double) (1.0 + ((double) (z * z)))))) * x))));
	}
	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

Original6.2
Target5.6
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.0165662251967751e30 or 7943.81812641865054 < y

    1. Initial program 3.7

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

    3. Applied associate-/r*1.2

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

    4. Simplified1.2

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

    6. Applied add-sqr-sqrt1.2

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

    7. Applied associate-/r*1.2

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

    if -4.0165662251967751e30 < y < 7943.81812641865054

    1. Initial program 8.8

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

    3. Applied div-inv8.8

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

    4. Applied associate-/l*9.0

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

    5. Simplified8.9

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.0165662251967751 \cdot 10^{30} \lor \neg \left(y \le 7943.81812641865054\right):\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{y}}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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)))))