Average Error: 6.4 → 4.0
Time: 3.5s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{x \cdot \left(y + z \cdot \left(y \cdot z\right)\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{x \cdot \left(y + z \cdot \left(y \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 (* x (+ y (* z (* y 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 / (x * (y + (z * (y * 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

Original6.4
Target5.8
Herbie4.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied clear-num_binary646.7

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

  4. Simplified6.7

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

  6. Applied distribute-rgt-in_binary646.7

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

  7. Simplified6.7

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

  8. Simplified6.7

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

  10. Applied associate-*r*_binary644.0

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

  11. Final simplification4.0

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

Reproduce

herbie shell --seed 2020273 
(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)))))