Average Error: 6.7 → 5.5
Time: 3.7s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -7.3367478733049452 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{x}{\frac{1}{1 + z \cdot z}}}\\ \mathbf{elif}\;\frac{1}{x} \le 1.1331728543600819 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \le -7.3367478733049452 \cdot 10^{72}:\\
\;\;\;\;\frac{\frac{1}{y}}{\frac{x}{\frac{1}{1 + z \cdot z}}}\\

\mathbf{elif}\;\frac{1}{x} \le 1.1331728543600819 \cdot 10^{81}:\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{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) (1.0 / x)) <= -7.336747873304945e+72)) {
		VAR = ((double) (((double) (1.0 / y)) / ((double) (x / ((double) (1.0 / ((double) (1.0 + ((double) (z * z))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / x)) <= 1.133172854360082e+81)) {
			VAR_1 = ((double) (((double) (((double) (1.0 / y)) / x)) / ((double) (1.0 + ((double) (z * z))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / y)) * ((double) (((double) (1.0 / x)) / ((double) (1.0 + ((double) (z * z))))))));
		}
		VAR = VAR_1;
	}
	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

Original6.7
Target6.0
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 x) < -7.3367478733049452e72

    1. Initial program 15.6

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

    3. Applied div-inv15.6

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

    4. Applied associate-/l*15.7

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

    5. Simplified15.6

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

    7. Applied associate-/r*15.6

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

    9. Applied *-un-lft-identity15.6

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

    10. Applied times-frac15.2

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

    11. Applied associate-/l*11.6

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

    if -7.3367478733049452e72 < (/ 1.0 x) < 1.1331728543600819e81

    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

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

    if 1.1331728543600819e81 < (/ 1.0 x)

    1. Initial program 15.6

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

    3. Applied *-un-lft-identity15.6

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

    4. Applied *-un-lft-identity15.6

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

    5. Applied times-frac15.6

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

    6. Applied times-frac12.2

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

    7. Simplified12.2

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -7.3367478733049452 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{x}{\frac{1}{1 + z \cdot z}}}\\ \mathbf{elif}\;\frac{1}{x} \le 1.1331728543600819 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))))