Average Error: 15.3 → 0.2
Time: 16.9s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) \cdot 2\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) \cdot 2\right) - z
double f(double x, double y, double z) {
        double r449155 = x;
        double r449156 = y;
        double r449157 = r449155 / r449156;
        double r449158 = log(r449157);
        double r449159 = r449155 * r449158;
        double r449160 = z;
        double r449161 = r449159 - r449160;
        return r449161;
}
↓
double f(double x, double y, double z) {
        double r449162 = x;
        double r449163 = cbrt(r449162);
        double r449164 = y;
        double r449165 = cbrt(r449164);
        double r449166 = r449163 / r449165;
        double r449167 = log(r449166);
        double r449168 = cbrt(r449165);
        double r449169 = r449168 * r449168;
        double r449170 = r449168 * r449169;
        double r449171 = r449163 / r449170;
        double r449172 = log(r449171);
        double r449173 = 2.0;
        double r449174 = r449172 * r449173;
        double r449175 = r449167 + r449174;
        double r449176 = r449162 * r449175;
        double r449177 = z;
        double r449178 = r449176 - r449177;
        return r449178;
}

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

Original15.3
Target7.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.3

    \[x \cdot \log \left(\frac{x}{y}\right) - z\]

  2. Simplified15.3

    \[\leadsto \color{blue}{\log \left(\frac{x}{y}\right) \cdot x - z}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt15.3

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

  5. Applied add-cube-cbrt15.3

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

  6. Applied times-frac15.3

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

  7. Applied log-prod3.6

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

  8. Simplified0.2

    \[\leadsto \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x - z\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))