Average Error: 6.4 → 6.2
Time: 4.1s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\frac{\frac{\frac{1}{1 + z \cdot z}}{x}}{y} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{1 + z \cdot z}}{x}}{y}

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ (/ 1.0 (+ 1.0 (* z z))) x) 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))) / 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

Original6.4
Target4.9
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.4

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

  3. Applied *-un-lft-identity_binary646.4

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

  4. Applied add-cube-cbrt_binary646.4

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot x}}{y \cdot \left(1 + z \cdot z\right)} \]

  5. Applied times-frac_binary646.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x}}}{y \cdot \left(1 + z \cdot z\right)} \]

  6. Applied times-frac_binary646.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}} \]

  7. Simplified6.3

    \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z} \]

  8. Simplified6.4

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

  10. Applied associate-*l/_binary646.4

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

  11. Simplified6.4

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

  13. Applied associate-/r*_binary646.2

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

  14. Simplified6.2

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

  15. Final simplification6.2

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

Reproduce

herbie shell --seed 2021198 
(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)))))