Average Error: 6.3 → 6.6
Time: 2.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{1 + z \cdot z} \cdot \frac{1}{x \cdot y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{1 + z \cdot z} \cdot \frac{1}{x \cdot 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 / (1.0 + (z * z))) * (1.0 / (x * 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

Original6.3
Target5.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied *-commutative6.3

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

  4. Applied div-inv6.3

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

  5. Applied times-frac6.5

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

  6. Simplified6.6

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

  7. Final simplification6.6

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

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)))))