Average Error: 15.2 → 1.3
Time: 9.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\left(\frac{\sqrt[3]{x}}{z} \cdot y\right) \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}}{\sqrt[3]{z}}}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\left(\frac{\sqrt[3]{x}}{z} \cdot y\right) \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}}{\sqrt[3]{z}}}{z + 1}
double f(double x, double y, double z) {
        double r265144 = x;
        double r265145 = y;
        double r265146 = r265144 * r265145;
        double r265147 = z;
        double r265148 = r265147 * r265147;
        double r265149 = 1.0;
        double r265150 = r265147 + r265149;
        double r265151 = r265148 * r265150;
        double r265152 = r265146 / r265151;
        return r265152;
}
↓
double f(double x, double y, double z) {
        double r265153 = x;
        double r265154 = cbrt(r265153);
        double r265155 = z;
        double r265156 = cbrt(r265155);
        double r265157 = r265156 * r265156;
        double r265158 = r265154 / r265157;
        double r265159 = r265154 / r265155;
        double r265160 = y;
        double r265161 = r265159 * r265160;
        double r265162 = cbrt(r265154);
        double r265163 = 3.0;
        double r265164 = pow(r265162, r265163);
        double r265165 = r265164 / r265156;
        double r265166 = r265161 * r265165;
        double r265167 = 1.0;
        double r265168 = r265155 + r265167;
        double r265169 = r265166 / r265168;
        double r265170 = r265158 * r265169;
        return r265170;
}

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

Original15.2
Target4.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.2

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

  3. Applied times-frac11.2

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

  5. Applied add-cube-cbrt11.6

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

  6. Applied times-frac6.6

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

  7. Applied associate-*l*1.3

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

  9. Applied add-cube-cbrt1.5

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

  11. Applied add-cube-cbrt1.6

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

  12. Applied times-frac1.6

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

  13. Applied associate-*l*1.4

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

  14. Simplified1.3

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

  15. Final simplification1.3

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))