Average Error: 6.8 → 0.2
Time: 14.4s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(-1 + z\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + y \cdot 0.3333333333333333\right)\right)\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(-1 + z\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + y \cdot 0.3333333333333333\right)\right)\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\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
 (-
  (-
   (*
    (+ -1.0 z)
    (-
     (* (pow y 4.0) -0.25)
     (+ y (* (* y y) (+ 0.5 (* y 0.3333333333333333))))))
   (* (+ -1.0 x) (log (/ 1.0 y))))
  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 (((-1.0 + z) * ((pow(y, 4.0) * -0.25) - (y + ((y * y) * (0.5 + (y * 0.3333333333333333)))))) - ((-1.0 + x) * log(1.0 / y))) - 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 6.8

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-\left(0.25 \cdot {y}^{4} + \left(y + \left(0.3333333333333333 \cdot {y}^{3} + 0.5 \cdot {y}^{2}\right)\right)\right)\right)}\right) - t\]

  3. Simplified0.2

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)}\right) - t\]

  4. Taylor expanded around inf 0.2

    \[\leadsto \left(\color{blue}{-1 \cdot \left(\left(x - 1\right) \cdot \log \left(\frac{1}{y}\right)\right)} + \left(z - 1\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)\right) - t\]

  5. Final simplification0.2

    \[\leadsto \left(\left(-1 + z\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + y \cdot 0.3333333333333333\right)\right)\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\right) - t\]

Reproduce

herbie shell --seed 2021044 
(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))