Average Error: 0.3 → 0.3
Time: 10.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\frac{\left(\log \left(x + y\right) + \log \left({z}^{\frac{2}{3}}\right)\right) \cdot \left(\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)\right)}{\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\frac{\left(\log \left(x + y\right) + \log \left({z}^{\frac{2}{3}}\right)\right) \cdot \left(\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)\right)}{\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t
double code(double x, double y, double z, double t, double a) {
	return (((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((((((log((x + y)) + log(pow(z, 0.6666666666666666))) * (log((x + y)) - log(pow(z, 0.6666666666666666)))) / (log((x + y)) - log(pow(z, 0.6666666666666666)))) + log(cbrt(z))) - t) + ((a - 0.5) * log(t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(\log \left(x + y\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  4. Applied log-prod0.3

    \[\leadsto \left(\left(\log \left(x + y\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  5. Applied associate-+r+0.3

    \[\leadsto \left(\color{blue}{\left(\left(\log \left(x + y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \log t\]
  6. Using strategy rm

  7. Applied flip-+0.3

    \[\leadsto \left(\left(\color{blue}{\frac{\log \left(x + y\right) \cdot \log \left(x + y\right) - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{\log \left(x + y\right) - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  8. Simplified0.3

    \[\leadsto \left(\left(\frac{\color{blue}{\left(\log \left(x + y\right) + \log \left({z}^{\frac{2}{3}}\right)\right) \cdot \left(\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)\right)}}{\log \left(x + y\right) - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  9. Simplified0.3

    \[\leadsto \left(\left(\frac{\left(\log \left(x + y\right) + \log \left({z}^{\frac{2}{3}}\right)\right) \cdot \left(\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)\right)}{\color{blue}{\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)}} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  10. Final simplification0.3

    \[\leadsto \left(\left(\frac{\left(\log \left(x + y\right) + \log \left({z}^{\frac{2}{3}}\right)\right) \cdot \left(\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)\right)}{\log \left(x + y\right) - \log \left({z}^{\frac{2}{3}}\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))