Average Error: 6.5 → 3.8
Time: 6.6s
Precision: binary64
[x y]: =sort([x y])
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ 1.0 y) (+ x (* z (* x 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) / (x + (z * (x * 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

Original6.5
Target5.1
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied *-un-lft-identity_binary64_109906.5

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

  4. Applied add-sqr-sqrt_binary64_110126.5

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

  5. Applied times-frac_binary64_109966.5

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

  6. Applied times-frac_binary64_109966.4

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

  7. Simplified6.4

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

  8. Simplified6.6

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

  10. Applied associate-*r*_binary64_109303.8

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

  12. Applied un-div-inv_binary64_109883.8

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

  13. Final simplification3.8

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

Reproduce

herbie shell --seed 2021015 
(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)))))