Average Error: 14.9 → 2.3
Time: 14.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{x}}{z} \cdot \left(\left(\sqrt[3]{x} \cdot \frac{y}{z}\right) \cdot \sqrt[3]{x}\right)}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{\sqrt[3]{x}}{z} \cdot \left(\left(\sqrt[3]{x} \cdot \frac{y}{z}\right) \cdot \sqrt[3]{x}\right)}{z + 1}
double f(double x, double y, double z) {
        double r185978 = x;
        double r185979 = y;
        double r185980 = r185978 * r185979;
        double r185981 = z;
        double r185982 = r185981 * r185981;
        double r185983 = 1.0;
        double r185984 = r185981 + r185983;
        double r185985 = r185982 * r185984;
        double r185986 = r185980 / r185985;
        return r185986;
}
↓
double f(double x, double y, double z) {
        double r185987 = x;
        double r185988 = cbrt(r185987);
        double r185989 = z;
        double r185990 = r185988 / r185989;
        double r185991 = y;
        double r185992 = r185991 / r185989;
        double r185993 = r185988 * r185992;
        double r185994 = r185993 * r185988;
        double r185995 = r185990 * r185994;
        double r185996 = 1.0;
        double r185997 = r185989 + r185996;
        double r185998 = r185995 / r185997;
        return r185998;
}

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

Original14.9
Target3.9
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.9

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

  3. Applied associate-/r*13.2

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

  4. Simplified2.6

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

  6. Applied *-un-lft-identity2.6

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

  7. Applied add-cube-cbrt3.1

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

  8. Applied times-frac3.1

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

  9. Applied associate-*r*2.3

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

  10. Simplified2.3

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

  11. Final simplification2.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))