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

Average Error: 15.3 → 0.3

Time: 22.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r22412387 = x;
        double r22412388 = y;
        double r22412389 = r22412387 / r22412388;
        double r22412390 = log(r22412389);
        double r22412391 = r22412387 * r22412390;
        double r22412392 = z;
        double r22412393 = r22412391 - r22412392;
        return r22412393;
}

↓

double f(double x, double y, double z) {
        double r22412394 = y;
        double r22412395 = cbrt(r22412394);
        double r22412396 = r22412395 * r22412395;
        double r22412397 = log(r22412396);
        double r22412398 = x;
        double r22412399 = -r22412398;
        double r22412400 = r22412397 * r22412399;
        double r22412401 = cbrt(r22412398);
        double r22412402 = cbrt(r22412395);
        double r22412403 = r22412401 / r22412402;
        double r22412404 = log(r22412403);
        double r22412405 = r22412403 * r22412403;
        double r22412406 = log(r22412405);
        double r22412407 = r22412404 + r22412406;
        double r22412408 = r22412407 * r22412398;
        double r22412409 = r22412400 + r22412408;
        double r22412410 = z;
        double r22412411 = r22412409 - r22412410;
        return r22412411;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.3
Target	7.6
Herbie	0.3

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

Derivation

1. Initial program 15.3

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

3. Applied add-cube-cbrt 15.3

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

4. Applied *-un-lft-identity 15.3

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

5. Applied times-frac 15.3

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

6. Applied log-prod 4.6

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

7. Applied distribute-rgt-in 4.6

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

8. Simplified 4.6

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

10. Applied add-cube-cbrt 4.6

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

11. Applied add-cube-cbrt 4.6

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

12. Applied times-frac 4.6

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

13. Applied log-prod 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019169 
(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))