Average Error: 6.1 → 5.6
Time: 7.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}
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 ((((cbrt(1.0) * cbrt(1.0)) / (cbrt(x) * cbrt(x))) / sqrt(fma(z, z, 1.0))) / (y / ((cbrt(1.0) / cbrt(x)) / sqrt(fma(z, z, 1.0)))));
}

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

Original6.1
Target5.5
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  2. Simplified6.3

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

  4. Applied add-sqr-sqrt6.3

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

  5. Applied add-cube-cbrt6.9

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

  6. Applied add-cube-cbrt6.9

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

  7. Applied times-frac6.9

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

  8. Applied times-frac6.9

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

  9. Applied associate-/l*5.6

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

  10. Final simplification5.6

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

Reproduce

herbie shell --seed 2020103 +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)))))