Average Error: 15.7 → 2.9
Time: 21.0s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 1.506248453983583083960974817287615558913 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{1}{z + 1}\right)\\ \mathbf{elif}\;z \le 80824642515684360464062931691110400:\\ \;\;\;\;\frac{x}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}} \cdot \frac{z}{\sqrt[3]{\frac{y}{z + 1}}}} \cdot \sqrt[3]{\frac{y}{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)} \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\right) \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le 1.506248453983583083960974817287615558913 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{z} \cdot \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{1}{z + 1}\right)\\

\mathbf{elif}\;z \le 80824642515684360464062931691110400:\\
\;\;\;\;\frac{x}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}} \cdot \frac{z}{\sqrt[3]{\frac{y}{z + 1}}}} \cdot \sqrt[3]{\frac{y}{z + 1}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)} \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\right) \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r61288934 = x;
        double r61288935 = y;
        double r61288936 = r61288934 * r61288935;
        double r61288937 = z;
        double r61288938 = r61288937 * r61288937;
        double r61288939 = 1.0;
        double r61288940 = r61288937 + r61288939;
        double r61288941 = r61288938 * r61288940;
        double r61288942 = r61288936 / r61288941;
        return r61288942;
}
↓
double f(double x, double y, double z) {
        double r61288943 = z;
        double r61288944 = 1.506248453983583e-68;
        bool r61288945 = r61288943 <= r61288944;
        double r61288946 = 1.0;
        double r61288947 = r61288946 / r61288943;
        double r61288948 = x;
        double r61288949 = r61288948 / r61288943;
        double r61288950 = y;
        double r61288951 = r61288949 * r61288950;
        double r61288952 = 1.0;
        double r61288953 = r61288943 + r61288952;
        double r61288954 = r61288946 / r61288953;
        double r61288955 = r61288951 * r61288954;
        double r61288956 = r61288947 * r61288955;
        double r61288957 = 8.082464251568436e+34;
        bool r61288958 = r61288943 <= r61288957;
        double r61288959 = r61288950 / r61288953;
        double r61288960 = cbrt(r61288959);
        double r61288961 = r61288943 / r61288960;
        double r61288962 = r61288961 * r61288961;
        double r61288963 = r61288948 / r61288962;
        double r61288964 = r61288963 * r61288960;
        double r61288965 = r61288949 * r61288959;
        double r61288966 = r61288947 * r61288965;
        double r61288967 = cbrt(r61288966);
        double r61288968 = r61288967 * r61288967;
        double r61288969 = r61288968 * r61288967;
        double r61288970 = r61288958 ? r61288964 : r61288969;
        double r61288971 = r61288945 ? r61288956 : r61288970;
        return r61288971;
}

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

Original15.7
Target4.1
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 1.506248453983583e-68

    1. Initial program 19.4

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

    3. Applied times-frac15.6

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

    5. Applied *-un-lft-identity15.6

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

    6. Applied times-frac8.5

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

    7. Applied associate-*l*3.2

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

    9. Applied div-inv3.2

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

    10. Applied associate-*r*3.9

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

    if 1.506248453983583e-68 < z < 8.082464251568436e+34

    1. Initial program 4.2

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

    3. Applied times-frac3.8

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

    5. Applied add-cube-cbrt4.6

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

    6. Applied associate-*r*4.6

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

    7. Simplified1.5

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

    if 8.082464251568436e+34 < z

    1. Initial program 11.6

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

    3. Applied times-frac4.7

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

    5. Applied *-un-lft-identity4.7

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

    6. Applied times-frac2.4

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

    7. Applied associate-*l*1.2

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

    9. Applied add-cube-cbrt1.5

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)} \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\right) \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1.506248453983583083960974817287615558913 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\left(\frac{x}{z} \cdot y\right) \cdot \frac{1}{z + 1}\right)\\ \mathbf{elif}\;z \le 80824642515684360464062931691110400:\\ \;\;\;\;\frac{x}{\frac{z}{\sqrt[3]{\frac{y}{z + 1}}} \cdot \frac{z}{\sqrt[3]{\frac{y}{z + 1}}}} \cdot \sqrt[3]{\frac{y}{z + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)} \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\right) \cdot \sqrt[3]{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))))