Average Error: 6.3 → 5.9
Time: 6.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x} \cdot \frac{1}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x} \cdot \frac{1}{\sqrt{1 + z \cdot z}}
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 / y) / sqrt((1.0 + (z * z)))) / x) * (1.0 / sqrt((1.0 + (z * z)))));
}

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

Original6.3
Target5.6
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied associate-/r*6.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}}\]

  4. Simplified6.5

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{1 + z \cdot z}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt6.5

    \[\leadsto \frac{\frac{\frac{1}{y}}{x}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}\]

  7. Applied div-inv6.6

    \[\leadsto \frac{\color{blue}{\frac{1}{y} \cdot \frac{1}{x}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}\]

  8. Applied times-frac6.0

    \[\leadsto \color{blue}{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}\]
  9. Using strategy rm

  10. Applied div-inv6.0

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

  11. Applied associate-*r*6.0

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

  12. Simplified5.9

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x}} \cdot \frac{1}{\sqrt{1 + z \cdot z}}\]

  13. Final simplification5.9

    \[\leadsto \frac{\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}}{x} \cdot \frac{1}{\sqrt{1 + z \cdot z}}\]

Reproduce

herbie shell --seed 2020078 
(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)))))