Average Error: 6.4 → 3.4
Time: 11.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right) + y \cdot 1}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right) + y \cdot 1}
double f(double x, double y, double z) {
        double r178515 = 1.0;
        double r178516 = x;
        double r178517 = r178515 / r178516;
        double r178518 = y;
        double r178519 = z;
        double r178520 = r178519 * r178519;
        double r178521 = r178515 + r178520;
        double r178522 = r178518 * r178521;
        double r178523 = r178517 / r178522;
        return r178523;
}
↓
double f(double x, double y, double z) {
        double r178524 = 1.0;
        double r178525 = x;
        double r178526 = r178524 / r178525;
        double r178527 = z;
        double r178528 = y;
        double r178529 = r178528 * r178527;
        double r178530 = r178527 * r178529;
        double r178531 = r178528 * r178524;
        double r178532 = r178530 + r178531;
        double r178533 = r178526 / r178532;
        return r178533;
}

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
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied distribute-rgt-in6.4

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

  4. Simplified3.4

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

  5. Final simplification3.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 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)))))