Average Error: 6.7 → 0.4
Time: 8.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\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
double f(double x, double y, double z, double t) {
        double r61232 = x;
        double r61233 = 1.0;
        double r61234 = r61232 - r61233;
        double r61235 = y;
        double r61236 = log(r61235);
        double r61237 = r61234 * r61236;
        double r61238 = z;
        double r61239 = r61238 - r61233;
        double r61240 = r61233 - r61235;
        double r61241 = log(r61240);
        double r61242 = r61239 * r61241;
        double r61243 = r61237 + r61242;
        double r61244 = t;
        double r61245 = r61243 - r61244;
        return r61245;
}
↓
double f(double x, double y, double z, double t) {
        double r61246 = y;
        double r61247 = cbrt(r61246);
        double r61248 = r61247 * r61247;
        double r61249 = log(r61248);
        double r61250 = x;
        double r61251 = 1.0;
        double r61252 = r61250 - r61251;
        double r61253 = r61249 * r61252;
        double r61254 = log(r61247);
        double r61255 = r61254 * r61252;
        double r61256 = z;
        double r61257 = r61256 - r61251;
        double r61258 = log(r61251);
        double r61259 = r61251 * r61246;
        double r61260 = 0.5;
        double r61261 = 2.0;
        double r61262 = pow(r61246, r61261);
        double r61263 = pow(r61251, r61261);
        double r61264 = r61262 / r61263;
        double r61265 = r61260 * r61264;
        double r61266 = r61259 + r61265;
        double r61267 = r61258 - r61266;
        double r61268 = r61257 * r61267;
        double r61269 = r61255 + r61268;
        double r61270 = r61253 + r61269;
        double r61271 = t;
        double r61272 = r61270 - r61271;
        return r61272;
}

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. 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-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020027 
(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))