Average Error: 7.3 → 0.4
Time: 10.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\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
double f(double x, double y, double z, double t) {
        double r74005 = x;
        double r74006 = 1.0;
        double r74007 = r74005 - r74006;
        double r74008 = y;
        double r74009 = log(r74008);
        double r74010 = r74007 * r74009;
        double r74011 = z;
        double r74012 = r74011 - r74006;
        double r74013 = r74006 - r74008;
        double r74014 = log(r74013);
        double r74015 = r74012 * r74014;
        double r74016 = r74010 + r74015;
        double r74017 = t;
        double r74018 = r74016 - r74017;
        return r74018;
}
↓
double f(double x, double y, double z, double t) {
        double r74019 = 2.0;
        double r74020 = y;
        double r74021 = cbrt(r74020);
        double r74022 = log(r74021);
        double r74023 = r74019 * r74022;
        double r74024 = x;
        double r74025 = 1.0;
        double r74026 = r74024 - r74025;
        double r74027 = r74023 * r74026;
        double r74028 = r74021 * r74021;
        double r74029 = cbrt(r74028);
        double r74030 = log(r74029);
        double r74031 = r74026 * r74030;
        double r74032 = cbrt(r74021);
        double r74033 = log(r74032);
        double r74034 = r74026 * r74033;
        double r74035 = r74031 + r74034;
        double r74036 = r74027 + r74035;
        double r74037 = z;
        double r74038 = r74037 - r74025;
        double r74039 = log(r74025);
        double r74040 = r74025 * r74020;
        double r74041 = 0.5;
        double r74042 = pow(r74020, r74019);
        double r74043 = pow(r74025, r74019);
        double r74044 = r74042 / r74043;
        double r74045 = r74041 * r74044;
        double r74046 = r74040 + r74045;
        double r74047 = r74039 - r74046;
        double r74048 = r74038 * r74047;
        double r74049 = r74036 + r74048;
        double r74050 = t;
        double r74051 = r74049 - r74050;
        return r74051;
}

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 7.3

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

  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

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

  9. Applied add-cube-cbrt0.4

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

  10. Applied cbrt-prod0.4

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

  11. Applied log-prod0.4

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

  12. Applied distribute-lft-in0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 
(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))