Average Error: 6.8 → 0.4
Time: 25.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(\log y \cdot \frac{1}{3}\right) \cdot \left(x - 1\right) + \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(\log y \cdot \frac{1}{3}\right) \cdot \left(x - 1\right) + \left(2 \cdot \log \left(\sqrt[3]{y}\right)\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
double f(double x, double y, double z, double t) {
        double r50032 = x;
        double r50033 = 1.0;
        double r50034 = r50032 - r50033;
        double r50035 = y;
        double r50036 = log(r50035);
        double r50037 = r50034 * r50036;
        double r50038 = z;
        double r50039 = r50038 - r50033;
        double r50040 = r50033 - r50035;
        double r50041 = log(r50040);
        double r50042 = r50039 * r50041;
        double r50043 = r50037 + r50042;
        double r50044 = t;
        double r50045 = r50043 - r50044;
        return r50045;
}
↓
double f(double x, double y, double z, double t) {
        double r50046 = y;
        double r50047 = log(r50046);
        double r50048 = 0.3333333333333333;
        double r50049 = r50047 * r50048;
        double r50050 = x;
        double r50051 = 1.0;
        double r50052 = r50050 - r50051;
        double r50053 = r50049 * r50052;
        double r50054 = 2.0;
        double r50055 = cbrt(r50046);
        double r50056 = log(r50055);
        double r50057 = r50054 * r50056;
        double r50058 = r50057 * r50052;
        double r50059 = r50053 + r50058;
        double r50060 = z;
        double r50061 = r50060 - r50051;
        double r50062 = log(r50051);
        double r50063 = r50051 * r50046;
        double r50064 = 0.5;
        double r50065 = pow(r50046, r50054);
        double r50066 = pow(r50051, r50054);
        double r50067 = r50065 / r50066;
        double r50068 = r50064 * r50067;
        double r50069 = r50063 + r50068;
        double r50070 = r50062 - r50069;
        double r50071 = r50061 * r50070;
        double r50072 = r50059 + r50071;
        double r50073 = t;
        double r50074 = r50072 - r50073;
        return r50074;
}

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 6.8

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

  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. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 
(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))