Average Error: 7.0 → 0.4
Time: 9.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + {\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right)\right)}^{1}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\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 \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + {\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right)\right)}^{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) - t
double f(double x, double y, double z, double t) {
        double r52164 = x;
        double r52165 = 1.0;
        double r52166 = r52164 - r52165;
        double r52167 = y;
        double r52168 = log(r52167);
        double r52169 = r52166 * r52168;
        double r52170 = z;
        double r52171 = r52170 - r52165;
        double r52172 = r52165 - r52167;
        double r52173 = log(r52172);
        double r52174 = r52171 * r52173;
        double r52175 = r52169 + r52174;
        double r52176 = t;
        double r52177 = r52175 - r52176;
        return r52177;
}
↓
double f(double x, double y, double z, double t) {
        double r52178 = x;
        double r52179 = 1.0;
        double r52180 = r52178 - r52179;
        double r52181 = 2.0;
        double r52182 = y;
        double r52183 = cbrt(r52182);
        double r52184 = log(r52183);
        double r52185 = r52181 * r52184;
        double r52186 = r52180 * r52185;
        double r52187 = 0.3333333333333333;
        double r52188 = pow(r52182, r52187);
        double r52189 = log(r52188);
        double r52190 = r52189 * r52180;
        double r52191 = 1.0;
        double r52192 = pow(r52190, r52191);
        double r52193 = r52186 + r52192;
        double r52194 = z;
        double r52195 = r52194 - r52179;
        double r52196 = log(r52179);
        double r52197 = r52179 * r52182;
        double r52198 = 0.5;
        double r52199 = pow(r52182, r52181);
        double r52200 = pow(r52179, r52181);
        double r52201 = r52199 / r52200;
        double r52202 = r52198 * r52201;
        double r52203 = r52197 + r52202;
        double r52204 = r52196 - r52203;
        double r52205 = r52195 * r52204;
        double r52206 = r52193 + r52205;
        double r52207 = t;
        double r52208 = r52206 - r52207;
        return r52208;
}

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 7.0

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

  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-lft-in0.4

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

  7. Simplified0.4

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

  9. Applied pow10.4

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

  10. Applied pow10.4

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

  11. Applied pow-prod-down0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020064 
(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))