Average Error: 0.1 → 0.1
Time: 4.1s
Precision: binary64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2} \cdot \sqrt[3]{\sqrt{y}}\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2} \cdot \sqrt[3]{\sqrt{y}}\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) - y\right) - z\right) + \log t
(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  (-
   (-
    (+
     (* x (log (* (pow (cbrt y) 2.0) (cbrt (sqrt y)))))
     (* x (log (cbrt (sqrt y)))))
    y)
   z)
  (log t)))
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (x * ((double) log(y)))) - y)) - z)) + ((double) log(t))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (((double) (x * ((double) log(((double) (((double) pow(((double) cbrt(y)), 2.0)) * ((double) cbrt(((double) sqrt(y)))))))))) + ((double) (x * ((double) log(((double) cbrt(((double) sqrt(y)))))))))) - y)) - z)) + ((double) 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 Error: 0.1 bits

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

  3. Applied add-cube-cbrtError: 0.1 bits

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

  4. Applied log-prodError: 0.1 bits

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

  5. Applied distribute-lft-inError: 0.1 bits

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

  6. SimplifiedError: 0.1 bits

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
  7. Using strategy rm

  8. Applied add-sqr-sqrtError: 0.1 bits

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

  9. Applied cbrt-prodError: 0.1 bits

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

  10. Applied log-prodError: 0.1 bits

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

  11. Applied distribute-lft-inError: 0.1 bits

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

  12. Applied associate-+r+Error: 0.1 bits

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

  13. SimplifiedError: 0.1 bits

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt{y}}\right)\right)} + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) - y\right) - z\right) + \log t\]
  14. Using strategy rm

  15. Applied add-log-expError: 0.1 bits

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

  16. Applied sum-logError: 0.1 bits

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

  17. SimplifiedError: 0.1 bits

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

  18. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020203 
(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)))