Average Error: 6.4 → 3.8
Time: 3.0s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.26703201546019796 \cdot 10^{63} \lor \neg \left(z \le 7.9906304383559719 \cdot 10^{105}\right):\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\right)} - \frac{1}{x \cdot {z}^{4}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.26703201546019796 \cdot 10^{63} \lor \neg \left(z \le 7.9906304383559719 \cdot 10^{105}\right):\\
\;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\right)} - \frac{1}{x \cdot {z}^{4}}}{y}\\

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

\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 (((z <= -1.267032015460198e+63) || !(z <= 7.990630438355972e+105))) {
		VAR = (((double) ((1.0 / ((double) (z * ((double) (z * x))))) - (1.0 / ((double) (x * ((double) pow(z, 4.0))))))) / y);
	} else {
		VAR = ((1.0 / x) / ((double) (((double) sqrt(((double) (1.0 + ((double) (z * z)))))) * ((double) (y * ((double) sqrt(((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
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.26703201546019796e63 or 7.9906304383559719e105 < z

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied *-un-lft-identity14.3

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

    5. Applied times-frac14.3

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

    6. Applied times-frac15.0

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

    7. Simplified15.0

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

    8. Simplified15.1

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

    10. Applied associate-*l/15.1

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

    11. Simplified15.1

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

    12. Taylor expanded around inf 15.1

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{x \cdot {z}^{2}} - 1 \cdot \frac{1}{x \cdot {z}^{4}}}}{y}\]

    13. Simplified7.7

      \[\leadsto \frac{\color{blue}{\frac{1}{z \cdot \left(z \cdot x\right)} - \frac{1}{x \cdot {z}^{4}}}}{y}\]

    if -1.26703201546019796e63 < z < 7.9906304383559719e105

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied associate-*r*1.2

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.26703201546019796 \cdot 10^{63} \lor \neg \left(z \le 7.9906304383559719 \cdot 10^{105}\right):\\ \;\;\;\;\frac{\frac{1}{z \cdot \left(z \cdot x\right)} - \frac{1}{x \cdot {z}^{4}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]

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)))))