Average Error: 6.9 → 0.4
Time: 24.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) + \left(\left(x - 1\right) \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) + 2 \cdot \log \left(\sqrt[3]{y}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) + \left(\left(x - 1\right) \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) + 2 \cdot \log \left(\sqrt[3]{y}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r56066 = x;
        double r56067 = 1.0;
        double r56068 = r56066 - r56067;
        double r56069 = y;
        double r56070 = log(r56069);
        double r56071 = r56068 * r56070;
        double r56072 = z;
        double r56073 = r56072 - r56067;
        double r56074 = r56067 - r56069;
        double r56075 = log(r56074);
        double r56076 = r56073 * r56075;
        double r56077 = r56071 + r56076;
        double r56078 = t;
        double r56079 = r56077 - r56078;
        return r56079;
}
↓
double f(double x, double y, double z, double t) {
        double r56080 = z;
        double r56081 = 1.0;
        double r56082 = r56080 - r56081;
        double r56083 = log(r56081);
        double r56084 = y;
        double r56085 = r56081 * r56084;
        double r56086 = 0.5;
        double r56087 = 2.0;
        double r56088 = pow(r56084, r56087);
        double r56089 = pow(r56081, r56087);
        double r56090 = r56088 / r56089;
        double r56091 = r56086 * r56090;
        double r56092 = r56085 + r56091;
        double r56093 = r56083 - r56092;
        double r56094 = r56082 * r56093;
        double r56095 = x;
        double r56096 = r56095 - r56081;
        double r56097 = 1.0;
        double r56098 = r56097 / r56084;
        double r56099 = -0.3333333333333333;
        double r56100 = pow(r56098, r56099);
        double r56101 = log(r56100);
        double r56102 = cbrt(r56084);
        double r56103 = log(r56102);
        double r56104 = r56087 * r56103;
        double r56105 = r56101 + r56104;
        double r56106 = r56096 * r56105;
        double r56107 = t;
        double r56108 = r56106 - r56107;
        double r56109 = r56094 + r56108;
        return r56109;
}

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 6.9

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

  2. Taylor expanded around 0 0.4

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

  4. Applied add-cube-cbrt0.4

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

  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-rgt-in0.5

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

  7. Applied associate-+l+0.5

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

  8. Taylor expanded around inf 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019294 
(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))