Average Error: 7.4 → 0.3
Time: 12.3s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 - y \cdot 0.3333333333333333\right)\right) - y\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 - y \cdot 0.3333333333333333\right)\right) - y\right)\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
 (-
  (+
   (* (log (sqrt y)) (- x 1.0))
   (+
    (* (log (sqrt y)) (- x 1.0))
    (* (- z 1.0) (- (* y (* y (- -0.5 (* y 0.3333333333333333)))) 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 ((log(sqrt(y)) * (x - 1.0)) + ((log(sqrt(y)) * (x - 1.0)) + ((z - 1.0) * ((y * (y * (-0.5 - (y * 0.3333333333333333)))) - 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 7.4

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

  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

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

  5. Applied add-sqr-sqrt_binary640.3

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

  6. Applied log-prod_binary640.3

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

  7. Applied distribute-rgt-in_binary640.3

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

  8. Applied associate-+l+_binary640.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020233 
(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))