Average Error: 6.9 → 0.4
Time: 5.7s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{1.6666666666666667} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\left(x - 1\right) \cdot \log \left({y}^{0.3333333333333333}\right) + \left(z - 1\right) \cdot \left(\log 1 + y \cdot \left(\frac{y}{1 \cdot 1} \cdot -0.5 - 1\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{1.6666666666666667} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\left(x - 1\right) \cdot \log \left({y}^{0.3333333333333333}\right) + \left(z - 1\right) \cdot \left(\log 1 + y \cdot \left(\frac{y}{1 \cdot 1} \cdot -0.5 - 1\right)\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) (log (* (pow (cbrt y) 1.6666666666666667) (cbrt (cbrt y)))))
   (+
    (* (- x 1.0) (log (pow y 0.3333333333333333)))
    (* (- z 1.0) (+ (log 1.0) (* y (- (* (/ y (* 1.0 1.0)) -0.5) 1.0))))))
  t))
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) log(y)))) + ((double) (((double) (z - 1.0)) * ((double) log(((double) (1.0 - y)))))))) - t));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (x - 1.0)) * ((double) log(((double) (((double) pow(((double) cbrt(y)), 1.6666666666666667)) * ((double) cbrt(((double) cbrt(y)))))))))) + ((double) (((double) (((double) (x - 1.0)) * ((double) log(((double) pow(y, 0.3333333333333333)))))) + ((double) (((double) (z - 1.0)) * ((double) (((double) log(1.0)) + ((double) (y * ((double) (((double) ((y / ((double) (1.0 * 1.0))) * -0.5)) - 1.0)))))))))))) - 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 Error: 6.9 bits

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

  2. Taylor expanded around 0 Error: 0.4 bits

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

  3. SimplifiedError: 0.4 bits

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

  5. Applied add-cube-cbrtError: 0.4 bits

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

  6. Applied log-prodError: 0.4 bits

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

  7. Applied distribute-lft-inError: 0.4 bits

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

  8. Applied associate-+l+Error: 0.4 bits

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

  9. SimplifiedError: 0.4 bits

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

  11. Applied pow1/3Error: 0.4 bits

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

  13. Applied add-cube-cbrtError: 0.4 bits

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

  14. Applied associate-*r*Error: 0.4 bits

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

  15. SimplifiedError: 0.4 bits

    \[\leadsto \left(\left(x - 1\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1.6666666666666667}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\left(x - 1\right) \cdot \log \left({y}^{0.3333333333333333}\right) + \left(z - 1\right) \cdot \left(\log 1 + y \cdot \left(\frac{y}{1 \cdot 1} \cdot -0.5 - 1\right)\right)\right)\right) - t\]

  16. Final simplificationError: 0.4 bits

    \[\leadsto \left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{1.6666666666666667} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\left(x - 1\right) \cdot \log \left({y}^{0.3333333333333333}\right) + \left(z - 1\right) \cdot \left(\log 1 + y \cdot \left(\frac{y}{1 \cdot 1} \cdot -0.5 - 1\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020203 
(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))