Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 14.6 → 0.2

Time: 25.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r32452150 = x;
        double r32452151 = y;
        double r32452152 = r32452150 / r32452151;
        double r32452153 = log(r32452152);
        double r32452154 = r32452150 * r32452153;
        double r32452155 = z;
        double r32452156 = r32452154 - r32452155;
        return r32452156;
}

↓

double f(double x, double y, double z) {
        double r32452157 = x;
        double r32452158 = cbrt(r32452157);
        double r32452159 = y;
        double r32452160 = cbrt(r32452159);
        double r32452161 = r32452158 / r32452160;
        double r32452162 = log(r32452161);
        double r32452163 = r32452157 * r32452162;
        double r32452164 = cbrt(r32452158);
        double r32452165 = cbrt(r32452160);
        double r32452166 = r32452164 / r32452165;
        double r32452167 = r32452166 * r32452166;
        double r32452168 = log(r32452167);
        double r32452169 = r32452167 * r32452167;
        double r32452170 = log(r32452169);
        double r32452171 = r32452168 + r32452170;
        double r32452172 = r32452171 * r32452157;
        double r32452173 = r32452163 + r32452172;
        double r32452174 = z;
        double r32452175 = r32452173 - r32452174;
        return r32452175;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.6
Target	7.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Initial program 14.6

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

3. Applied add-cube-cbrt 14.6

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

4. Applied add-cube-cbrt 14.6

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

5. Applied times-frac 14.6

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

6. Applied log-prod 3.3

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

7. Applied distribute-lft-in 3.3

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

8. Simplified 3.3

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

10. Applied add-cube-cbrt 3.3

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

11. Applied add-cube-cbrt 3.3

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

12. Applied times-frac 3.3

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

13. Applied add-cube-cbrt 3.3

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

14. Applied add-cube-cbrt 3.3

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

15. Applied times-frac 3.3

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

16. Applied swap-sqr 3.3

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

17. Applied log-prod 0.2

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

18. Simplified 0.2

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

19. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019158 
(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))