Average Error: 6.8 → 0.4
Time: 5.4s
Precision: binary64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(z - 1\right) \cdot \left(\log 1 + \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot -0.5 - 1 \cdot y\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(z - 1\right) \cdot \left(\log 1 + \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot -0.5 - 1 \cdot y\right)\right)\right) - 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) (((double) (x - 1.0)) * ((double) (2.0 * ((double) (((double) log(((double) cbrt(y)))) + ((double) log(((double) cbrt(((double) cbrt(y)))))))))))) + ((double) (((double) (x - 1.0)) * ((double) log(((double) cbrt(((double) cbrt(y)))))))))) + ((double) (((double) (z - 1.0)) * ((double) (((double) log(1.0)) + ((double) (((double) (((double) ((y / 1.0) * (y / 1.0))) * -0.5)) - ((double) (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 0.4

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

  3. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  10. Applied add-cube-cbrt0.4

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

  11. Applied log-prod0.4

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

  12. Applied distribute-lft-in0.4

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

  13. Applied associate-+r+0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020199 
(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))