Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x + \left(\left(\log t - y\right) - z\right)\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x + \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x + \left(\left(\log t - y\right) - z\right)\right)
double code(double x, double y, double z, double t) {
	return ((((x * log(y)) - y) - z) + log(t));
}
↓
double code(double x, double y, double z, double t) {
	return ((log((cbrt(y) * cbrt(y))) * x) + ((log(pow((1.0 / y), -0.3333333333333333)) * x) + ((log(t) - y) - z)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
  2. Using strategy rm

  3. Applied sub-neg0.1

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

  4. Applied associate--l+0.1

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

  5. Applied associate-+l+0.1

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

  6. Simplified0.1

    \[\leadsto x \cdot \log y + \color{blue}{\left(\left(\log t - y\right) - z\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.1

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

  9. Applied log-prod0.1

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

  10. Applied distribute-rgt-in0.1

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

  11. Applied associate-+l+0.1

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

  12. Taylor expanded around inf 0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020058 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))