Average Error: 6.6 → 0.4
Time: 9.5s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r59118 = x;
        double r59119 = 1.0;
        double r59120 = r59118 - r59119;
        double r59121 = y;
        double r59122 = log(r59121);
        double r59123 = r59120 * r59122;
        double r59124 = z;
        double r59125 = r59124 - r59119;
        double r59126 = r59119 - r59121;
        double r59127 = log(r59126);
        double r59128 = r59125 * r59127;
        double r59129 = r59123 + r59128;
        double r59130 = t;
        double r59131 = r59129 - r59130;
        return r59131;
}
↓
double f(double x, double y, double z, double t) {
        double r59132 = 1.0;
        double r59133 = y;
        double r59134 = r59132 / r59133;
        double r59135 = -0.3333333333333333;
        double r59136 = pow(r59134, r59135);
        double r59137 = cbrt(r59133);
        double r59138 = r59136 * r59137;
        double r59139 = log(r59138);
        double r59140 = x;
        double r59141 = 1.0;
        double r59142 = r59140 - r59141;
        double r59143 = r59139 * r59142;
        double r59144 = log(r59137);
        double r59145 = r59144 * r59142;
        double r59146 = z;
        double r59147 = r59146 - r59141;
        double r59148 = log(r59141);
        double r59149 = r59141 * r59133;
        double r59150 = 0.5;
        double r59151 = 2.0;
        double r59152 = pow(r59133, r59151);
        double r59153 = pow(r59141, r59151);
        double r59154 = r59152 / r59153;
        double r59155 = r59150 * r59154;
        double r59156 = r59149 + r59155;
        double r59157 = r59148 - r59156;
        double r59158 = r59147 * r59157;
        double r59159 = r59145 + r59158;
        double r59160 = r59143 + r59159;
        double r59161 = t;
        double r59162 = r59160 - r59161;
        return r59162;
}

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 6.6

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.4

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

  6. Applied distribute-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Taylor expanded around inf 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020039 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))