Average Error: 7.0 → 0.5
Time: 7.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(0.5, {y}^{2}, -\mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(0.5, {y}^{2}, -\mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t\right)
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 fma(log(y), (x - 1.0), (fma(0.5, pow(y, 2.0), -fma(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. Simplified7.0

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

  3. Taylor expanded around 0 0.4

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

  4. Taylor expanded around inf 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(0.5, {y}^{2}, -\mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))