Average Error: 6.3 → 5.0
Time: 3.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \le -1.19195819535175784 \cdot 10^{32} \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 7.24453765483583405 \cdot 10^{88}\right):\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + z \cdot z}}}{y \cdot \left(\sqrt{1 + z \cdot z} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \le -1.19195819535175784 \cdot 10^{32} \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 7.24453765483583405 \cdot 10^{88}\right):\\
\;\;\;\;\frac{\frac{1}{\sqrt{1 + z \cdot z}}}{y \cdot \left(\sqrt{1 + z \cdot z} \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{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) (y * ((double) (1.0 + ((double) (z * z)))))) <= -1.1919581953517578e+32) || !(((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= 7.244537654835834e+88))) {
		VAR = ((double) (((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) (z * z)))))))) / ((double) (y * ((double) (((double) sqrt(((double) (1.0 + ((double) (z * z)))))) * x))))));
	} else {
		VAR = ((double) (((double) (1.0 / x)) / ((double) (((double) (y * ((double) sqrt(((double) (1.0 + ((double) (z * z)))))))) * ((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

Original6.3
Target5.5
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 (+ 1.0 (* z z))) < -1.1919581953517578e+32 or 7.244537654835834e+88 < (* y (+ 1.0 (* z z)))

    1. Initial program 9.4

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

    3. Applied *-un-lft-identity9.4

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

    4. Applied *-un-lft-identity9.4

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

    5. Applied times-frac9.4

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

    6. Applied times-frac7.5

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

    7. Simplified7.5

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

    9. Applied associate-*l/7.4

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

    10. Simplified7.4

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

    12. Applied add-sqr-sqrt7.4

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

    13. Applied div-inv7.4

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

    14. Applied times-frac7.4

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

    15. Applied associate-/l*7.4

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

    16. Simplified7.4

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

    if -1.1919581953517578e+32 < (* y (+ 1.0 (* z z))) < 7.244537654835834e+88

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied associate-*r*0.3

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \le -1.19195819535175784 \cdot 10^{32} \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \le 7.24453765483583405 \cdot 10^{88}\right):\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + z \cdot z}}}{y \cdot \left(\sqrt{1 + z \cdot z} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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)))))