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

Average Error: 14.8 → 0.3

Time: 11.6s

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 r7869186 = x;
        double r7869187 = y;
        double r7869188 = r7869186 / r7869187;
        double r7869189 = log(r7869188);
        double r7869190 = r7869186 * r7869189;
        double r7869191 = z;
        double r7869192 = r7869190 - r7869191;
        return r7869192;
}

↓

double f(double x, double y, double z) {
        double r7869193 = x;
        double r7869194 = cbrt(r7869193);
        double r7869195 = r7869194 * r7869194;
        double r7869196 = log(r7869195);
        double r7869197 = r7869196 * r7869193;
        double r7869198 = cbrt(r7869194);
        double r7869199 = y;
        double r7869200 = cbrt(r7869199);
        double r7869201 = r7869198 / r7869200;
        double r7869202 = log(r7869201);
        double r7869203 = r7869202 + r7869202;
        double r7869204 = r7869203 + r7869202;
        double r7869205 = r7869204 * r7869193;
        double r7869206 = z;
        double r7869207 = r7869205 - r7869206;
        double r7869208 = r7869197 + r7869207;
        return r7869208;
}

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 +o rules:numerics
(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))