Average Error: 0.1 → 0.1
Time: 7.3s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(x \cdot \log \left({y}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot x - y\right)\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(x \cdot \log \left({y}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot x - y\right)\right) - z\right) + \log t
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 ((((x * log(pow(y, 0.6666666666666666))) + ((log(cbrt(y)) * x) - y)) - z) + log(t));
}

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 add-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-rgt-in0.1

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

  6. Applied associate--l+0.1

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

  8. Applied *-un-lft-identity0.1

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

  9. Applied cbrt-prod0.1

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

  10. Applied associate-*l*0.1

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

  11. Simplified0.1

    \[\leadsto \left(\left(\log \left(\sqrt[3]{1} \cdot \color{blue}{{y}^{\frac{2}{3}}}\right) \cdot x + \left(\log \left(\sqrt[3]{y}\right) \cdot x - y\right)\right) - z\right) + \log t\]

  12. Taylor expanded around 0 0.1

    \[\leadsto \left(\left(\color{blue}{x \cdot \log \left({y}^{\frac{2}{3}}\right)} + \left(\log \left(\sqrt[3]{y}\right) \cdot x - y\right)\right) - z\right) + \log t\]

  13. Final simplification0.1

    \[\leadsto \left(\left(x \cdot \log \left({y}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot x - y\right)\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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)))