Average Error: 6.7 → 5.6
Time: 3.3s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 2.526817269739846 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{y}}{1 + z \cdot z}\\ \end{array}\]
\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) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 2.526817269739846 \cdot 10^{+96}\right):\\
\;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{y}}{1 + z \cdot z}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y (+ 1.0 (* z z))) (- INFINITY))
         (not (<= (* y (+ 1.0 (* z z))) 2.526817269739846e+96)))
   (/ (/ -1.0 (* y x)) (- -1.0 (* z z)))
   (* (/ 1.0 x) (/ (/ 1.0 y) (+ 1.0 (* z z))))))
double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (((y * (1.0 + (z * z))) <= -((double) INFINITY)) || !((y * (1.0 + (z * z))) <= 2.526817269739846e+96)) {
		tmp = (-1.0 / (y * x)) / (-1.0 - (z * z));
	} else {
		tmp = (1.0 / x) * ((1.0 / y) / (1.0 + (z * z)));
	}
	return tmp;
}

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
Target5.9
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 y (+.f64 1 (*.f64 z z))) < -inf.0 or 2.5268172697398459e96 < (*.f64 y (+.f64 1 (*.f64 z z)))

    1. Initial program 13.7

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

    3. Applied associate-/r*_binary6411.3

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

    5. Applied frac-2neg_binary6411.3

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

    6. Simplified11.4

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

    7. Simplified11.4

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

    if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 2.5268172697398459e96

    1. Initial program 0.3

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

    3. Applied associate-/r*_binary642.6

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

    5. Applied *-un-lft-identity_binary642.6

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

    6. Applied div-inv_binary642.7

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

    7. Applied times-frac_binary640.4

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

    8. Simplified0.4

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 2.526817269739846 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{y}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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)))))