Average Error: 7.0 → 1.1
Time: 8.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) \cdot \sqrt[3]{\log y} + \left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) \cdot \sqrt[3]{\log y} + \left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)\right) - t
double code(double x, double y, double z, double t) {
	return ((((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t);
}
↓
double code(double x, double y, double z, double t) {
	return (((((x - 1.0) * (cbrt(log(y)) * cbrt(log(y)))) * cbrt(log(y))) + ((0.5 * pow(y, 2.0)) - ((1.0 * (z * y)) + (0.5 * (z * pow(y, 2.0)))))) - 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 7.0

    \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

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

  3. Taylor expanded around inf 0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.1

    \[\leadsto \left(\left(x - 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right) \cdot \sqrt[3]{\log y}\right)} + \left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)\right) - t\]

  6. Applied associate-*r*1.1

    \[\leadsto \left(\color{blue}{\left(\left(x - 1\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) \cdot \sqrt[3]{\log y}} + \left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)\right) - t\]

  7. Final simplification1.1

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) \cdot \sqrt[3]{\log y} + \left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))