Average Error: 7.1 → 0.6
Time: 26.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(y \cdot \left(0.5 \cdot y\right) - \left(y \cdot \left(0.5 \cdot y\right)\right) \cdot z\right) - \left(1 \cdot z\right) \cdot y\right) + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(y \cdot \left(0.5 \cdot y\right) - \left(y \cdot \left(0.5 \cdot y\right)\right) \cdot z\right) - \left(1 \cdot z\right) \cdot y\right) + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1892989 = x;
        double r1892990 = 1.0;
        double r1892991 = r1892989 - r1892990;
        double r1892992 = y;
        double r1892993 = log(r1892992);
        double r1892994 = r1892991 * r1892993;
        double r1892995 = z;
        double r1892996 = r1892995 - r1892990;
        double r1892997 = r1892990 - r1892992;
        double r1892998 = log(r1892997);
        double r1892999 = r1892996 * r1892998;
        double r1893000 = r1892994 + r1892999;
        double r1893001 = t;
        double r1893002 = r1893000 - r1893001;
        return r1893002;
}
↓
double f(double x, double y, double z, double t) {
        double r1893003 = y;
        double r1893004 = 0.5;
        double r1893005 = r1893004 * r1893003;
        double r1893006 = r1893003 * r1893005;
        double r1893007 = z;
        double r1893008 = r1893006 * r1893007;
        double r1893009 = r1893006 - r1893008;
        double r1893010 = 1.0;
        double r1893011 = r1893010 * r1893007;
        double r1893012 = r1893011 * r1893003;
        double r1893013 = r1893009 - r1893012;
        double r1893014 = cbrt(r1893003);
        double r1893015 = log(r1893014);
        double r1893016 = x;
        double r1893017 = r1893016 - r1893010;
        double r1893018 = r1893015 * r1893017;
        double r1893019 = r1893018 + r1893018;
        double r1893020 = r1893019 + r1893018;
        double r1893021 = r1893013 + r1893020;
        double r1893022 = t;
        double r1893023 = r1893021 - r1893022;
        return r1893023;
}

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 0.4

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

  3. Simplified0.4

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

  4. Taylor expanded around inf 0.5

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

  5. Simplified0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied log-prod0.6

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

  9. Applied distribute-rgt-in0.6

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 
(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))