Average Error: 7.6 → 0.3
Time: 4.4s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right) \cdot \left(1 - z\right)\right) - t\]
\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 + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right) \cdot \left(1 - z\right)\right) - t
(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))
↓
(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (* (+ y (* (* y y) (+ (* y 0.3333333333333333) 0.5))) (- 1.0 z)))
  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) * log(y)) + ((y + ((y * y) * ((y * 0.3333333333333333) + 0.5))) * (1.0 - 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 7.6

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-\left(0.3333333333333333 \cdot {y}^{3} + \left(0.5 \cdot {y}^{2} + y\right)\right)\right)}\right) - t\]

  3. Simplified0.3

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(-y\right) - \left(y \cdot y\right) \cdot \left(0.3333333333333333 \cdot y + 0.5\right)\right)}\right) - t\]

  4. Final simplification0.3

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right) \cdot \left(1 - z\right)\right) - t\]

Reproduce

herbie shell --seed 2020232 
(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))