Average Error: 14.8 → 2.3
Time: 6.3s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{1} \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z + 1}\right)}} \cdot \frac{\sqrt[3]{x}}{{\left(\sqrt[3]{z}\right)}^{3}}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{1} \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z + 1}\right)}} \cdot \frac{\sqrt[3]{x}}{{\left(\sqrt[3]{z}\right)}^{3}}
double f(double x, double y, double z) {
        double r285877 = x;
        double r285878 = y;
        double r285879 = r285877 * r285878;
        double r285880 = z;
        double r285881 = r285880 * r285880;
        double r285882 = 1.0;
        double r285883 = r285880 + r285882;
        double r285884 = r285881 * r285883;
        double r285885 = r285879 / r285884;
        return r285885;
}
↓
double f(double x, double y, double z) {
        double r285886 = x;
        double r285887 = cbrt(r285886);
        double r285888 = z;
        double r285889 = 1.0;
        double r285890 = cbrt(r285889);
        double r285891 = y;
        double r285892 = 1.0;
        double r285893 = r285888 + r285892;
        double r285894 = r285891 / r285893;
        double r285895 = r285887 * r285894;
        double r285896 = r285890 * r285895;
        double r285897 = r285888 / r285896;
        double r285898 = r285887 / r285897;
        double r285899 = cbrt(r285888);
        double r285900 = 3.0;
        double r285901 = pow(r285899, r285900);
        double r285902 = r285887 / r285901;
        double r285903 = r285898 * r285902;
        return r285903;
}

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

Original14.8
Target4.1
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 14.8

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

  3. Applied times-frac10.9

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

  5. Applied add-cube-cbrt11.3

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

  6. Applied times-frac6.4

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

  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

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

  10. Applied *-un-lft-identity1.5

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

  11. Applied cbrt-prod1.5

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

  12. Applied times-frac1.5

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

  13. Applied associate-*l*1.6

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

  14. Final simplification2.3

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

Reproduce

herbie shell --seed 1978988140 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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