Average Error: 7.3 → 0.4
Time: 10.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{1} \cdot {y}^{\frac{2}{3}}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\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]{1} \cdot {y}^{\frac{2}{3}}\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
double f(double x, double y, double z, double t) {
        double r64024 = x;
        double r64025 = 1.0;
        double r64026 = r64024 - r64025;
        double r64027 = y;
        double r64028 = log(r64027);
        double r64029 = r64026 * r64028;
        double r64030 = z;
        double r64031 = r64030 - r64025;
        double r64032 = r64025 - r64027;
        double r64033 = log(r64032);
        double r64034 = r64031 * r64033;
        double r64035 = r64029 + r64034;
        double r64036 = t;
        double r64037 = r64035 - r64036;
        return r64037;
}
↓
double f(double x, double y, double z, double t) {
        double r64038 = x;
        double r64039 = 1.0;
        double r64040 = r64038 - r64039;
        double r64041 = 1.0;
        double r64042 = cbrt(r64041);
        double r64043 = y;
        double r64044 = 0.6666666666666666;
        double r64045 = pow(r64043, r64044);
        double r64046 = r64042 * r64045;
        double r64047 = log(r64046);
        double r64048 = r64040 * r64047;
        double r64049 = cbrt(r64043);
        double r64050 = log(r64049);
        double r64051 = r64040 * r64050;
        double r64052 = r64048 + r64051;
        double r64053 = z;
        double r64054 = r64053 - r64039;
        double r64055 = log(r64039);
        double r64056 = r64039 * r64043;
        double r64057 = 0.5;
        double r64058 = 2.0;
        double r64059 = pow(r64043, r64058);
        double r64060 = pow(r64039, r64058);
        double r64061 = r64059 / r64060;
        double r64062 = r64057 * r64061;
        double r64063 = r64056 + r64062;
        double r64064 = r64055 - r64063;
        double r64065 = r64054 * r64064;
        double r64066 = r64052 + r64065;
        double r64067 = t;
        double r64068 = r64066 - r64067;
        return r64068;
}

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. Using strategy rm

  8. Applied *-un-lft-identity0.4

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

  9. Applied cbrt-prod0.4

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

  10. Applied associate-*l*0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))