Average Error: 15.4 → 1.5
Time: 46.7s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
↓
\[\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}
↓
\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)
double f(double x, double y, double z) {
        double r11300387 = x;
        double r11300388 = y;
        double r11300389 = r11300387 * r11300388;
        double r11300390 = z;
        double r11300391 = r11300390 * r11300390;
        double r11300392 = 1.0;
        double r11300393 = r11300390 + r11300392;
        double r11300394 = r11300391 * r11300393;
        double r11300395 = r11300389 / r11300394;
        return r11300395;
}
↓
double f(double x, double y, double z) {
        double r11300396 = x;
        double r11300397 = cbrt(r11300396);
        double r11300398 = r11300397 * r11300397;
        double r11300399 = cbrt(r11300398);
        double r11300400 = r11300399 * r11300399;
        double r11300401 = r11300399 * r11300400;
        double r11300402 = z;
        double r11300403 = r11300401 / r11300402;
        double r11300404 = y;
        double r11300405 = 1.0;
        double r11300406 = r11300405 + r11300402;
        double r11300407 = r11300404 / r11300406;
        double r11300408 = r11300397 / r11300402;
        double r11300409 = r11300407 * r11300408;
        double r11300410 = r11300403 * r11300409;
        return r11300410;
}

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

Original15.4
Target4.3
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1.0 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.4

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac11.2

    \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1.0}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt11.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{z \cdot z} \cdot \frac{y}{z + 1.0}\]

  6. Applied times-frac6.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1.0}\]

  7. Applied associate-*l*1.3

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  10. Final simplification1.5

    \[\leadsto \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)\]

Reproduce

herbie shell --seed 2019165 +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))))