Average Error: 7.3 → 0.4
Time: 34.2s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(\left(\sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + 1 \cdot y} + \sqrt{\log 1}\right) \cdot \left(z - 1\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + 1 \cdot y}\right) + \log \left(\sqrt{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt{y}\right) \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(\left(\sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + 1 \cdot y} + \sqrt{\log 1}\right) \cdot \left(z - 1\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + 1 \cdot y}\right) + \log \left(\sqrt{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt{y}\right) \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2369783 = x;
        double r2369784 = 1.0;
        double r2369785 = r2369783 - r2369784;
        double r2369786 = y;
        double r2369787 = log(r2369786);
        double r2369788 = r2369785 * r2369787;
        double r2369789 = z;
        double r2369790 = r2369789 - r2369784;
        double r2369791 = r2369784 - r2369786;
        double r2369792 = log(r2369791);
        double r2369793 = r2369790 * r2369792;
        double r2369794 = r2369788 + r2369793;
        double r2369795 = t;
        double r2369796 = r2369794 - r2369795;
        return r2369796;
}
↓
double f(double x, double y, double z, double t) {
        double r2369797 = 0.5;
        double r2369798 = y;
        double r2369799 = 1.0;
        double r2369800 = r2369798 / r2369799;
        double r2369801 = r2369800 * r2369800;
        double r2369802 = r2369797 * r2369801;
        double r2369803 = r2369799 * r2369798;
        double r2369804 = r2369802 + r2369803;
        double r2369805 = sqrt(r2369804);
        double r2369806 = log(r2369799);
        double r2369807 = sqrt(r2369806);
        double r2369808 = r2369805 + r2369807;
        double r2369809 = z;
        double r2369810 = r2369809 - r2369799;
        double r2369811 = r2369808 * r2369810;
        double r2369812 = r2369807 - r2369805;
        double r2369813 = r2369811 * r2369812;
        double r2369814 = sqrt(r2369798);
        double r2369815 = log(r2369814);
        double r2369816 = x;
        double r2369817 = r2369816 - r2369799;
        double r2369818 = r2369815 * r2369817;
        double r2369819 = r2369813 + r2369818;
        double r2369820 = r2369819 + r2369818;
        double r2369821 = t;
        double r2369822 = r2369820 - r2369821;
        return r2369822;
}

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 7.3

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

  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. Simplified0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied log-prod0.3

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

  7. Applied distribute-rgt-in0.3

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

  8. Applied associate-+l+0.3

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied difference-of-squares0.4

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

  13. Applied associate-*r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))