Average Error: 7.1 → 0.9
Time: 6.5s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left({\left(\sqrt[3]{{\log y}^{2}}\right)}^{0.6666666666666666} \cdot \sqrt[3]{\sqrt[3]{{\log y}^{2}}}\right)\right) \cdot \sqrt[3]{\log y} + \left(y + y \cdot \left(y \cdot 0.5 - z\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \left({\left(\sqrt[3]{{\log y}^{2}}\right)}^{0.6666666666666666} \cdot \sqrt[3]{\sqrt[3]{{\log y}^{2}}}\right)\right) \cdot \sqrt[3]{\log y} + \left(y + y \cdot \left(y \cdot 0.5 - z\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
 (-
  (+
   (*
    (*
     (- x 1.0)
     (*
      (pow (cbrt (pow (log y) 2.0)) 0.6666666666666666)
      (cbrt (cbrt (pow (log y) 2.0)))))
    (cbrt (log y)))
   (+ y (* y (- (* y 0.5) 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) * (pow(cbrt(pow(log(y), 2.0)), 0.6666666666666666) * cbrt(cbrt(pow(log(y), 2.0))))) * cbrt(log(y))) + (y + (y * ((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 7.1

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied add-cube-cbrt_binary641.1

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

  6. Applied associate-*r*_binary641.1

    \[\leadsto \left(\color{blue}{\left(\left(x - 1\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) \cdot \sqrt[3]{\log y}} + \left(y + y \cdot \left(0.5 \cdot y - z\right)\right)\right) - t\]
  7. Using strategy rm

  8. Applied pow1/3_binary6464.0

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

  9. Applied pow1/3_binary6464.0

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

  10. Applied pow-prod-down_binary641.0

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

  11. Simplified1.0

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

  13. Applied add-cube-cbrt_binary641.1

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

  14. Simplified1.0

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{{\log y}^{2}}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{{\left({\log y}^{2}\right)}^{0.3333333333333333}}\right)\right) \cdot \sqrt[3]{\log y} + \left(y + y \cdot \left(0.5 \cdot y - z\right)\right)\right) - t\]

  15. Simplified0.9

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left({\left(\sqrt[3]{{\log y}^{2}}\right)}^{0.6666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{{\log y}^{2}}}}\right)\right) \cdot \sqrt[3]{\log y} + \left(y + y \cdot \left(0.5 \cdot y - z\right)\right)\right) - t\]

  16. Final simplification0.9

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left({\left(\sqrt[3]{{\log y}^{2}}\right)}^{0.6666666666666666} \cdot \sqrt[3]{\sqrt[3]{{\log y}^{2}}}\right)\right) \cdot \sqrt[3]{\log y} + \left(y + y \cdot \left(y \cdot 0.5 - z\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020273 
(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))