Average Error: 7.0 → 0.4
Time: 9.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{4}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \left(\log \left(\sqrt{\sqrt{y}}\right) + \log \left(\sqrt{y}\right)\right) + \left(\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) \cdot \left(z - 1\right) - t\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{4}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \left(\log \left(\sqrt{\sqrt{y}}\right) + \log \left(\sqrt{y}\right)\right) + \left(\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) \cdot \left(z - 1\right) - t\right)\right)
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) log(((double) pow(((double) (1.0 / y)), -0.25)))) * ((double) (x - 1.0)))) + ((double) (((double) (((double) (x - 1.0)) * ((double) (((double) log(((double) sqrt(((double) sqrt(y)))))) + ((double) log(((double) sqrt(y)))))))) + ((double) (((double) (((double) (((double) log(1.0)) - ((double) (((double) (1.0 * y)) + ((double) (0.5 * ((double) (((double) pow(y, 2.0)) / ((double) pow(1.0, 2.0)))))))))) * ((double) (z - 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 7.0

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

  2. Taylor expanded around 0 0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied distribute-lft-in0.3

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

  7. Applied associate-+l+0.3

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

  8. Simplified0.3

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

  10. Applied add-sqr-sqrt0.3

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

  11. Applied sqrt-prod0.3

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

  12. Applied log-prod0.3

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

  13. Applied distribute-rgt-in0.3

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

  14. Applied associate-+l+0.3

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

  15. Applied associate--l+0.3

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

  16. Simplified0.4

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

  17. Taylor expanded around inf 0.4

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

  18. Final simplification0.4

    \[\leadsto \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{4}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \left(\log \left(\sqrt{\sqrt{y}}\right) + \log \left(\sqrt{y}\right)\right) + \left(\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) \cdot \left(z - 1\right) - t\right)\right)\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))