Average Error: 6.0 → 5.4
Time: 31.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{y}
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) sqrt(1.0)) / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) / ((double) (((double) cbrt(((double) fma(z, z, 1.0)))) * ((double) cbrt(((double) fma(z, z, 1.0)))))))) * ((double) (((double) (((double) (((double) sqrt(1.0)) / ((double) cbrt(x)))) / ((double) cbrt(((double) fma(z, z, 1.0)))))) / 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

Original6.0
Target5.3
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.0

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

  2. Simplified5.8

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

  4. Applied *-un-lft-identity5.8

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

  5. Applied add-cube-cbrt5.9

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

  6. Applied add-cube-cbrt6.4

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

  7. Applied add-sqr-sqrt6.4

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

  8. Applied times-frac6.4

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

  9. Applied times-frac6.4

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

  10. Applied times-frac5.4

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

  11. Simplified5.4

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

  12. Final simplification5.4

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))