Average Error: 5.7 → 1.5
Time: 4.3s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{1}{y}}\\ \frac{t_0 \cdot t_0}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(1, z\right) \cdot x} \end{array} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\frac{1}{y}}\\
\frac{t_0 \cdot t_0}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(1, z\right) \cdot x}
\end{array}

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 y))))
   (* (/ (* t_0 t_0) (hypot 1.0 z)) (/ t_0 (* (hypot 1.0 z) x)))))
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 t_0 = cbrt(1.0 / y);
	return ((t_0 * t_0) / hypot(1.0, z)) * (t_0 / (hypot(1.0, z) * x));
}

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

Original5.7
Target4.4
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 5.7

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

  2. Simplified5.7

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

  3. Applied *-un-lft-identity_binary645.7

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

  4. Applied add-cube-cbrt_binary645.7

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot x}}{y \cdot \mathsf{fma}\left(z, z, 1\right)} \]

  5. Applied times-frac_binary645.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x}}}{y \cdot \mathsf{fma}\left(z, z, 1\right)} \]

  6. Applied times-frac_binary645.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\mathsf{fma}\left(z, z, 1\right)}} \]

  7. Simplified5.6

    \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\mathsf{fma}\left(z, z, 1\right)} \]

  8. Simplified5.7

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

  9. Applied un-div-inv_binary645.6

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

  10. Applied add-sqr-sqrt_binary645.6

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

  11. Applied associate-*l*_binary645.6

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

  12. Simplified5.6

    \[\leadsto \frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \color{blue}{\left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]

  13. Applied add-cube-cbrt_binary646.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot \sqrt[3]{\frac{1}{y}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)} \]

  14. Applied times-frac_binary645.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\sqrt[3]{\frac{1}{y}}}{x \cdot \mathsf{hypot}\left(1, z\right)}} \]

  15. Simplified1.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\sqrt[3]{\frac{1}{y}}}{x \cdot \mathsf{hypot}\left(1, z\right)} \]

  16. Final simplification1.5

    \[\leadsto \frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\sqrt[3]{\frac{1}{y}}}{\mathsf{hypot}\left(1, z\right) \cdot x} \]

Reproduce

herbie shell --seed 2022088 
(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)))))