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

Average Error: 15.2 → 0.2

Time: 7.4s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) log(((double) (x / y)))))) - z));
}

↓

double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) (((double) log(((double) (((double) sqrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) / ((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))))))) + ((double) log(((double) (((double) sqrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))))))))) + ((double) (((double) (((double) log(((double) (((double) cbrt(x)) / ((double) cbrt(y)))))) * x)) - z))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.2
Target	7.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Initial program 15.2

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

3. Applied add-cube-cbrt 15.2

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

4. Applied add-cube-cbrt 15.2

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

5. Applied times-frac 15.2

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

6. Applied log-prod 3.4

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

7. Applied distribute-lft-in 3.4

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

8. Applied associate--l+ 3.4

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

9. Simplified 3.4

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

11. Applied *-un-lft-identity 3.4

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

12. Applied cbrt-prod 3.4

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

13. Applied *-un-lft-identity 3.4

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

14. Applied cbrt-prod 3.4

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

15. Applied swap-sqr 3.4

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

16. Applied add-sqr-sqrt 3.4

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

17. Applied times-frac 3.4

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

18. Applied log-prod 0.2

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

19. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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