Average Error: 6.2 → 1.7
Time: 3.3s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{y \cdot \left(1 \cdot x\right) + \left(y \cdot z\right) \cdot \left(x \cdot z\right)}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{y \cdot \left(1 \cdot x\right) + \left(y \cdot z\right) \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 (* 1.0 x)) (* (* y z) (* x z)))))
double code(double x, double y, double z) {
	return ((1.0 / x) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))));
}
↓
double code(double x, double y, double z) {
	return (1.0 / ((double) (((double) (y * ((double) (1.0 * x)))) + ((double) (((double) (y * z)) * ((double) (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

Original6.2
Target5.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program Error: 6.2 bits

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

  2. SimplifiedError: 6.4 bits

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

  4. Applied distribute-lft-inError: 6.4 bits

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

  5. Applied distribute-lft-inError: 6.4 bits

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

  6. SimplifiedError: 6.4 bits

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

  7. SimplifiedError: 3.6 bits

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

  9. Applied associate-*r*Error: 1.7 bits

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

  10. Final simplificationError: 1.7 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))))