Average Error: 6.7 → 0.4
Time: 28.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(y \cdot 1 + \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right)\right)\right) + \left(x - 1\right) \cdot \log \left({y}^{\frac{1}{3}} \cdot \sqrt[3]{y}\right)\right) - t\]
\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 \log \left(\sqrt[3]{y}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(y \cdot 1 + \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right)\right)\right) + \left(x - 1\right) \cdot \log \left({y}^{\frac{1}{3}} \cdot \sqrt[3]{y}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2394128 = x;
        double r2394129 = 1.0;
        double r2394130 = r2394128 - r2394129;
        double r2394131 = y;
        double r2394132 = log(r2394131);
        double r2394133 = r2394130 * r2394132;
        double r2394134 = z;
        double r2394135 = r2394134 - r2394129;
        double r2394136 = r2394129 - r2394131;
        double r2394137 = log(r2394136);
        double r2394138 = r2394135 * r2394137;
        double r2394139 = r2394133 + r2394138;
        double r2394140 = t;
        double r2394141 = r2394139 - r2394140;
        return r2394141;
}
↓
double f(double x, double y, double z, double t) {
        double r2394142 = x;
        double r2394143 = 1.0;
        double r2394144 = r2394142 - r2394143;
        double r2394145 = y;
        double r2394146 = cbrt(r2394145);
        double r2394147 = log(r2394146);
        double r2394148 = r2394144 * r2394147;
        double r2394149 = z;
        double r2394150 = r2394149 - r2394143;
        double r2394151 = log(r2394143);
        double r2394152 = r2394145 * r2394143;
        double r2394153 = r2394145 / r2394143;
        double r2394154 = r2394153 * r2394153;
        double r2394155 = 0.5;
        double r2394156 = r2394154 * r2394155;
        double r2394157 = r2394152 + r2394156;
        double r2394158 = r2394151 - r2394157;
        double r2394159 = r2394150 * r2394158;
        double r2394160 = r2394148 + r2394159;
        double r2394161 = 0.3333333333333333;
        double r2394162 = pow(r2394145, r2394161);
        double r2394163 = r2394162 * r2394146;
        double r2394164 = log(r2394163);
        double r2394165 = r2394144 * r2394164;
        double r2394166 = r2394160 + r2394165;
        double r2394167 = t;
        double r2394168 = r2394166 - r2394167;
        return r2394168;
}

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 6.7

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

  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

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  10. Applied pow1/30.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(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))