Average Error: 6.3 → 5.9
Time: 11.6s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.92819790221646427 \cdot 10^{23}:\\ \;\;\;\;\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.92819790221646427 \cdot 10^{23}:\\
\;\;\;\;\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\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 ((x <= -1.9281979022164643e+23)) {
		VAR = ((double) (((double) (1.0 / ((double) (x * y)))) / ((double) (1.0 + ((double) (z * z))))));
	} else {
		VAR = ((double) (((double) (1.0 / y)) / ((double) (((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

Original6.3
Target5.7
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.9281979022164643e+23

    1. Initial program 1.5

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

    3. Applied associate-/r*1.4

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

    4. Simplified1.4

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

    6. Applied div-inv1.4

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

    7. Applied associate-/l*1.8

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

    8. Simplified1.8

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

    if -1.9281979022164643e+23 < x

    1. Initial program 8.0

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

    3. Applied associate-/r*8.6

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

    4. Simplified8.6

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

    6. Applied div-inv8.6

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

    7. Applied associate-/l*7.4

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

    8. Simplified7.3

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.92819790221646427 \cdot 10^{23}:\\ \;\;\;\;\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

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)))))