Average Error: 6.5 → 3.9
Time: 4.6s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\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
Target5.9
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

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_binary646.5

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

  4. Applied add-cube-cbrt_binary646.5

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

  5. Applied times-frac_binary646.5

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

  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

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

  9. Taylor expanded around 0 6.6

    \[\leadsto \color{blue}{\frac{1}{y \cdot \left(x + x \cdot {z}^{2}\right)}}\]

  10. Simplified6.6

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

  12. Applied add-sqr-sqrt_binary646.6

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

  13. Applied associate-/l*_binary646.6

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

  14. Simplified6.6

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

  16. Applied associate-*r*_binary643.9

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

  17. Final simplification3.9

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

Reproduce

herbie shell --seed 2021173 
(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)))))