Average Error: 6.4 → 3.4
Time: 4.2s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)} \]
\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

Original6.4
Target5.7
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.4

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

  3. Applied associate-/r*_binary646.5

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

  4. Simplified6.7

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

  6. Applied *-un-lft-identity_binary646.7

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

  7. Applied times-frac_binary646.5

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

  8. Applied associate-/l*_binary646.4

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

  9. Simplified6.3

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

  11. Applied associate-*l*_binary643.4

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

  12. Final simplification3.4

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

Reproduce

herbie shell --seed 2021204 
(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)))))