Average Error: 6.6 → 1.6
Time: 10.3s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\frac{\frac{\frac{-1}{x}}{-\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot y} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{-1}{x}}{-\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right) \cdot y}

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (/ (/ (/ -1.0 x) (- (hypot 1.0 z))) (* (hypot 1.0 z) y)))
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) {
	return ((-1.0 / x) / -hypot(1.0, z)) / (hypot(1.0, z) * y);
}

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

Original6.6
Target5.1
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.6

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

  2. Simplified6.6

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

  3. Applied add-cube-cbrt_binary647.2

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

  4. Applied times-frac_binary646.8

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

  5. Applied add-sqr-sqrt_binary646.8

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

  6. Applied *-un-lft-identity_binary646.8

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

  7. Applied add-cube-cbrt_binary646.8

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

  8. Applied times-frac_binary646.8

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

  9. Applied cbrt-prod_binary646.8

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

  10. Applied times-frac_binary646.8

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

  11. Simplified6.8

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

  12. Simplified5.9

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

  13. Applied associate-*l/_binary645.9

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

  14. Applied frac-times_binary642.2

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

  15. Simplified1.6

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

  16. Applied frac-2neg_binary641.6

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

  17. Simplified1.6

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

  18. Final simplification1.6

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

Reproduce

herbie shell --seed 2021280 
(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)))))