Average Error: 6.7 → 0.4
Time: 31.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + \left(\log 1 - 1 \cdot y\right)\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + \left(\log 1 - 1 \cdot y\right)\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2248231 = x;
        double r2248232 = 1.0;
        double r2248233 = r2248231 - r2248232;
        double r2248234 = y;
        double r2248235 = log(r2248234);
        double r2248236 = r2248233 * r2248235;
        double r2248237 = z;
        double r2248238 = r2248237 - r2248232;
        double r2248239 = r2248232 - r2248234;
        double r2248240 = log(r2248239);
        double r2248241 = r2248238 * r2248240;
        double r2248242 = r2248236 + r2248241;
        double r2248243 = t;
        double r2248244 = r2248242 - r2248243;
        return r2248244;
}
↓
double f(double x, double y, double z, double t) {
        double r2248245 = y;
        double r2248246 = 0.3333333333333333;
        double r2248247 = pow(r2248245, r2248246);
        double r2248248 = log(r2248247);
        double r2248249 = x;
        double r2248250 = 1.0;
        double r2248251 = r2248249 - r2248250;
        double r2248252 = r2248248 * r2248251;
        double r2248253 = z;
        double r2248254 = r2248253 - r2248250;
        double r2248255 = -0.5;
        double r2248256 = r2248245 / r2248250;
        double r2248257 = r2248256 * r2248256;
        double r2248258 = r2248255 * r2248257;
        double r2248259 = log(r2248250);
        double r2248260 = r2248250 * r2248245;
        double r2248261 = r2248259 - r2248260;
        double r2248262 = r2248258 + r2248261;
        double r2248263 = r2248254 * r2248262;
        double r2248264 = r2248252 + r2248263;
        double r2248265 = cbrt(r2248245);
        double r2248266 = r2248265 * r2248265;
        double r2248267 = log(r2248266);
        double r2248268 = r2248267 * r2248251;
        double r2248269 = r2248264 + r2248268;
        double r2248270 = t;
        double r2248271 = r2248269 - r2248270;
        return r2248271;
}

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 6.7

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Applied associate-+l+0.4

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

  10. Applied pow1/30.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))