Average Error: 6.7 → 2.1
Time: 13.9s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(1, z\right)}\\ \frac{\frac{1}{\sqrt[3]{y}}}{t_0} \cdot \frac{\frac{\frac{1}{\sqrt[3]{y} \cdot x}}{\mathsf{hypot}\left(1, z\right)}}{\sqrt[3]{y} \cdot t_0} \end{array} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{hypot}\left(1, z\right)}\\
\frac{\frac{1}{\sqrt[3]{y}}}{t_0} \cdot \frac{\frac{\frac{1}{\sqrt[3]{y} \cdot x}}{\mathsf{hypot}\left(1, z\right)}}{\sqrt[3]{y} \cdot t_0}
\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 (sqrt (hypot 1.0 z))))
   (*
    (/ (/ 1.0 (cbrt y)) t_0)
    (/ (/ (/ 1.0 (* (cbrt y) x)) (hypot 1.0 z)) (* (cbrt y) t_0)))))
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 = sqrt(hypot(1.0, z));
	return ((1.0 / cbrt(y)) / t_0) * (((1.0 / (cbrt(y) * x)) / hypot(1.0, z)) / (cbrt(y) * t_0));
}

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
Target5.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.7

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

  2. Simplified6.7

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

  3. Applied associate-/r*_binary647.0

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

  4. Applied add-sqr-sqrt_binary647.0

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

  5. Applied add-cube-cbrt_binary647.6

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

  6. Applied *-un-lft-identity_binary647.6

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

  7. Applied add-cube-cbrt_binary647.6

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

  8. Applied times-frac_binary647.6

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

  9. Applied times-frac_binary647.6

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

  10. Applied times-frac_binary646.5

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

  11. Simplified6.5

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

  12. Simplified2.2

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

  13. Applied add-sqr-sqrt_binary642.2

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

  14. Applied *-un-lft-identity_binary642.2

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

  15. Applied cbrt-prod_binary642.2

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

  16. Applied add-cube-cbrt_binary642.4

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

  17. Applied add-cube-cbrt_binary642.4

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

  18. Applied times-frac_binary642.4

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

  19. Applied times-frac_binary642.4

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

  20. Applied times-frac_binary641.3

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

  21. Simplified1.3

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

  22. Simplified1.3

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

  23. Applied add-sqr-sqrt_binary641.3

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

  24. Applied *-un-lft-identity_binary641.3

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

  25. Applied times-frac_binary641.3

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

  26. Applied times-frac_binary641.3

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

  27. Applied associate-*l*_binary641.3

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

  28. Simplified2.1

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

  29. Final simplification2.1

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

Reproduce

herbie shell --seed 2021275 
(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)))))