Average Error: 6.3 → 3.6
Time: 3.4s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y + z \cdot \left(y \cdot z\right)}}{x}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ 1.0 (+ y (* z (* y 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 + (z * (y * 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

Original6.3
Target5.6
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied distribute-rgt-in_binary64_106936.3

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

  4. Simplified6.3

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

  5. Simplified6.3

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

  7. Applied associate-*r*_binary64_106833.6

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

  9. Applied div-inv_binary64_107403.7

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

  10. Simplified3.7

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

  12. Applied associate-*l/_binary64_106863.6

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

  13. Simplified3.6

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

  14. Final simplification3.6

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

Reproduce

herbie shell --seed 2020289 
(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)))))