Average Error: 7.2 → 0.4
Time: 20.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \frac{-2}{3}\right) \cdot \log \left(\frac{1}{y}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \frac{-2}{3}\right) \cdot \log \left(\frac{1}{y}\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 r57973 = x;
        double r57974 = 1.0;
        double r57975 = r57973 - r57974;
        double r57976 = y;
        double r57977 = log(r57976);
        double r57978 = r57975 * r57977;
        double r57979 = z;
        double r57980 = r57979 - r57974;
        double r57981 = r57974 - r57976;
        double r57982 = log(r57981);
        double r57983 = r57980 * r57982;
        double r57984 = r57978 + r57983;
        double r57985 = t;
        double r57986 = r57984 - r57985;
        return r57986;
}
↓
double f(double x, double y, double z, double t) {
        double r57987 = x;
        double r57988 = 1.0;
        double r57989 = r57987 - r57988;
        double r57990 = -0.6666666666666666;
        double r57991 = r57989 * r57990;
        double r57992 = 1.0;
        double r57993 = y;
        double r57994 = r57992 / r57993;
        double r57995 = log(r57994);
        double r57996 = r57991 * r57995;
        double r57997 = cbrt(r57993);
        double r57998 = log(r57997);
        double r57999 = r57998 * r57989;
        double r58000 = z;
        double r58001 = r58000 - r57988;
        double r58002 = log(r57988);
        double r58003 = r57988 * r57993;
        double r58004 = 0.5;
        double r58005 = 2.0;
        double r58006 = pow(r57993, r58005);
        double r58007 = pow(r57988, r58005);
        double r58008 = r58006 / r58007;
        double r58009 = r58004 * r58008;
        double r58010 = r58003 + r58009;
        double r58011 = r58002 - r58010;
        double r58012 = r58001 * r58011;
        double r58013 = r57999 + r58012;
        double r58014 = r57996 + r58013;
        double r58015 = t;
        double r58016 = r58014 - r58015;
        return r58016;
}

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 7.2

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

  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. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around inf 0.4

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

  11. Applied log-pow0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 
(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))