Average Error: 16.1 → 0.5
Time: 7.0s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 8.742822810316433167447774358180572573041 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{{\left(-1 \cdot y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{{\left(\frac{1}{y}\right)}^{\frac{-1}{3}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\\ \end{array}\]
x \cdot \log \left(\frac{x}{y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le 8.742822810316433167447774358180572573041 \cdot 10^{-311}:\\
\;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{{\left(-1 \cdot y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{{\left(\frac{1}{y}\right)}^{\frac{-1}{3}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r585891 = x;
        double r585892 = y;
        double r585893 = r585891 / r585892;
        double r585894 = log(r585893);
        double r585895 = r585891 * r585894;
        return r585895;
}
↓
double f(double x, double y) {
        double r585896 = y;
        double r585897 = 8.7428228103164e-311;
        bool r585898 = r585896 <= r585897;
        double r585899 = x;
        double r585900 = 2.0;
        double r585901 = cbrt(r585899);
        double r585902 = cbrt(r585896);
        double r585903 = r585901 / r585902;
        double r585904 = log(r585903);
        double r585905 = r585900 * r585904;
        double r585906 = -1.0;
        double r585907 = r585906 * r585896;
        double r585908 = 0.3333333333333333;
        double r585909 = pow(r585907, r585908);
        double r585910 = cbrt(r585906);
        double r585911 = r585909 * r585910;
        double r585912 = r585901 / r585911;
        double r585913 = log(r585912);
        double r585914 = r585905 + r585913;
        double r585915 = r585899 * r585914;
        double r585916 = 1.0;
        double r585917 = r585916 / r585896;
        double r585918 = -0.3333333333333333;
        double r585919 = pow(r585917, r585918);
        double r585920 = r585901 / r585919;
        double r585921 = log(r585920);
        double r585922 = r585900 * r585921;
        double r585923 = r585922 + r585904;
        double r585924 = r585899 * r585923;
        double r585925 = r585898 ? r585915 : r585924;
        return r585925;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.1
Target8.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt 1.297314905261780286102316169197038563602 \cdot 10^{-303}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\log x - \log y}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 8.7428228103164e-311

    1. Initial program 15.7

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

    3. Applied add-cube-cbrt15.7

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

    4. Applied add-cube-cbrt15.7

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

    5. Applied times-frac15.7

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

    6. Applied log-prod3.9

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

    7. Simplified0.4

      \[\leadsto x \cdot \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]

    8. Taylor expanded around -inf 0.5

      \[\leadsto x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\color{blue}{{\left(-1 \cdot y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}}\right)\right)\]

    if 8.7428228103164e-311 < y

    1. Initial program 16.4

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

    3. Applied add-cube-cbrt16.4

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

    4. Applied add-cube-cbrt16.4

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

    5. Applied times-frac16.4

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

    6. Applied log-prod4.5

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

    7. Simplified0.4

      \[\leadsto x \cdot \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]

    8. Taylor expanded around inf 0.6

      \[\leadsto x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\color{blue}{{\left(\frac{1}{y}\right)}^{\frac{-1}{3}}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 8.742822810316433167447774358180572573041 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{{\left(-1 \cdot y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{{\left(\frac{1}{y}\right)}^{\frac{-1}{3}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(FPCore (x y)
  :name "Data.HyperLogLog.Type:size from hyperloglog-0.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y 1.29731490526178e-303) (* x (log (/ x y))) (/ x (/ 1 (- (log x) (log y)))))

  (* x (log (/ x y))))