Average Error: 7.0 → 0.5
Time: 8.1s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(\sqrt[3]{z - 1} \cdot \sqrt[3]{z - 1}\right) \cdot \left(\sqrt[3]{z - 1} \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x - 1\right) \cdot \log y + \left(\sqrt[3]{z - 1} \cdot \sqrt[3]{z - 1}\right) \cdot \left(\sqrt[3]{z - 1} \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) log(y)))) + ((double) (((double) (z - 1.0)) * ((double) log(((double) (1.0 - y)))))))) - t));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) log(y)))) + ((double) (((double) (((double) cbrt(((double) (z - 1.0)))) * ((double) cbrt(((double) (z - 1.0)))))) * ((double) (((double) cbrt(((double) (z - 1.0)))) * ((double) (((double) log(1.0)) - ((double) (((double) (1.0 * y)) + ((double) (0.5 * ((double) (((double) pow(y, 2.0)) / ((double) pow(1.0, 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 0.3

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied associate-*l*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020130 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))