Average Error: 6.8 → 0.5
Time: 27.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(z - 1\right) \cdot \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{1} \cdot \frac{y \cdot y}{1}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(z - 1\right) \cdot \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{1} \cdot \frac{y \cdot y}{1}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r58144 = x;
        double r58145 = 1.0;
        double r58146 = r58144 - r58145;
        double r58147 = y;
        double r58148 = log(r58147);
        double r58149 = r58146 * r58148;
        double r58150 = z;
        double r58151 = r58150 - r58145;
        double r58152 = r58145 - r58147;
        double r58153 = log(r58152);
        double r58154 = r58151 * r58153;
        double r58155 = r58149 + r58154;
        double r58156 = t;
        double r58157 = r58155 - r58156;
        return r58157;
}
↓
double f(double x, double y, double z, double t) {
        double r58158 = z;
        double r58159 = 1.0;
        double r58160 = r58158 - r58159;
        double r58161 = log(r58159);
        double r58162 = y;
        double r58163 = r58159 * r58162;
        double r58164 = r58161 - r58163;
        double r58165 = 0.5;
        double r58166 = r58165 / r58159;
        double r58167 = r58162 * r58162;
        double r58168 = r58167 / r58159;
        double r58169 = r58166 * r58168;
        double r58170 = r58164 - r58169;
        double r58171 = r58160 * r58170;
        double r58172 = x;
        double r58173 = r58172 - r58159;
        double r58174 = cbrt(r58162);
        double r58175 = log(r58174);
        double r58176 = r58173 * r58175;
        double r58177 = r58174 * r58174;
        double r58178 = log(r58177);
        double r58179 = r58178 * r58173;
        double r58180 = r58176 + r58179;
        double r58181 = r58171 + r58180;
        double r58182 = t;
        double r58183 = r58181 - r58182;
        return r58183;
}

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

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

  3. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.5

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

  7. Applied distribute-lft-in0.5

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 
(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))