Average Error: 6.4 → 6.3
Time: 2.8s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ (/ 1.0 y) (+ 1.0 (* z z))) x))
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) / (1.0 + (z * z))) / x;
}

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

Original6.4
Target5.7
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied div-inv_binary646.4

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

  5. Applied associate-*l/_binary646.4

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

  6. Simplified6.4

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

  8. Applied associate-/r*_binary646.3

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

  9. Final simplification6.3

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

Reproduce

herbie shell --seed 2020232 
(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)))))