Average Error: 6.2 → 2.4
Time: 5.3s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 1.0543199916138207 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 1.0543199916138207 \cdot 10^{+73}:\\
\;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{x}}{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 (<= y 1.0543199916138207e+73)
   (/ (/ 1.0 (+ y (* z (* y z)))) x)
   (/ (/ (/ 1.0 x) 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 <= 1.0543199916138207e+73) {
		tmp = (1.0 / (y + (z * (y * z)))) / x;
	} else {
		tmp = ((1.0 / x) / 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

Original6.2
Target4.9
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1.05431999161382075e73

    1. Initial program 7.4

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

    3. Applied div-inv_binary647.4

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

    4. Simplified7.4

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

    6. Applied associate-*l/_binary647.4

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

    7. Simplified7.4

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

    9. Applied associate-*l*_binary643.2

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

    if 1.05431999161382075e73 < y

    1. Initial program 4.4

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

    3. Applied associate-/r*_binary641.1

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.0543199916138207 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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)))))