Average Error: 6.6 → 0.5
Time: 7.3s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(y \cdot \left(y \cdot 0.5\right) - \left(1 \cdot \left(y \cdot z\right) + \left(y \cdot \left(y \cdot 0.5\right)\right) \cdot z\right)\right)\right) - t\]
\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(y \cdot \left(y \cdot 0.5\right) - \left(1 \cdot \left(y \cdot z\right) + \left(y \cdot \left(y \cdot 0.5\right)\right) \cdot z\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) (y * ((double) (y * 0.5)))) - ((double) (((double) (1.0 * ((double) (y * z)))) + ((double) (((double) (y * ((double) (y * 0.5)))) * z)))))))) - 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: 6.6 bits

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

  2. Taylor expanded around 0 Error: 0.4 bits

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

  3. SimplifiedError: 0.4 bits

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

  4. Taylor expanded around inf Error: 0.5 bits

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

  5. SimplifiedError: 0.5 bits

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

  6. Final simplificationError: 0.5 bits

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(y \cdot \left(y \cdot 0.5\right) - \left(1 \cdot \left(y \cdot z\right) + \left(y \cdot \left(y \cdot 0.5\right)\right) \cdot z\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020200 
(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))