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

Average Error: 14.8 → 0.3

Time: 11.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r9258588 = x;
        double r9258589 = y;
        double r9258590 = r9258588 / r9258589;
        double r9258591 = log(r9258590);
        double r9258592 = r9258588 * r9258591;
        double r9258593 = z;
        double r9258594 = r9258592 - r9258593;
        return r9258594;
}

↓

double f(double x, double y, double z) {
        double r9258595 = x;
        double r9258596 = cbrt(r9258595);
        double r9258597 = r9258596 * r9258596;
        double r9258598 = log(r9258597);
        double r9258599 = r9258598 * r9258595;
        double r9258600 = cbrt(r9258596);
        double r9258601 = y;
        double r9258602 = cbrt(r9258601);
        double r9258603 = r9258600 / r9258602;
        double r9258604 = log(r9258603);
        double r9258605 = r9258604 + r9258604;
        double r9258606 = r9258605 + r9258604;
        double r9258607 = r9258606 * r9258595;
        double r9258608 = z;
        double r9258609 = r9258607 - r9258608;
        double r9258610 = r9258599 + r9258609;
        return r9258610;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.8
Target	7.7
Herbie	0.3

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

Derivation

1. Initial program 14.8

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

3. Applied *-un-lft-identity 14.8

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

4. Applied add-cube-cbrt 14.8

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

5. Applied times-frac 14.8

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

6. Applied log-prod 4.7

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

7. Applied distribute-rgt-in 4.7

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

8. Applied associate--l+ 4.7

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

10. Applied add-cube-cbrt 4.7

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

11. Applied add-cube-cbrt 4.7

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

12. Applied times-frac 4.7

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

13. Applied log-prod 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019156 
(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))