Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Average Error: 6.8 → 0.4

Time: 30.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r47150 = x;
        double r47151 = 1.0;
        double r47152 = r47150 - r47151;
        double r47153 = y;
        double r47154 = log(r47153);
        double r47155 = r47152 * r47154;
        double r47156 = z;
        double r47157 = r47156 - r47151;
        double r47158 = r47151 - r47153;
        double r47159 = log(r47158);
        double r47160 = r47157 * r47159;
        double r47161 = r47155 + r47160;
        double r47162 = t;
        double r47163 = r47161 - r47162;
        return r47163;
}

↓

double f(double x, double y, double z, double t) {
        double r47164 = y;
        double r47165 = cbrt(r47164);
        double r47166 = r47165 * r47165;
        double r47167 = log(r47166);
        double r47168 = x;
        double r47169 = 1.0;
        double r47170 = r47168 - r47169;
        double r47171 = r47167 * r47170;
        double r47172 = log(r47164);
        double r47173 = exp(r47172);
        double r47174 = cbrt(r47173);
        double r47175 = log(r47174);
        double r47176 = r47170 * r47175;
        double r47177 = r47171 + r47176;
        double r47178 = z;
        double r47179 = r47178 - r47169;
        double r47180 = log(r47169);
        double r47181 = 0.5;
        double r47182 = 2.0;
        double r47183 = pow(r47164, r47182);
        double r47184 = pow(r47169, r47182);
        double r47185 = r47183 / r47184;
        double r47186 = r47181 * r47185;
        double r47187 = fma(r47169, r47164, r47186);
        double r47188 = r47180 - r47187;
        double r47189 = r47179 * r47188;
        double r47190 = r47177 + r47189;
        double r47191 = t;
        double r47192 = r47190 - r47191;
        return r47192;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

10. Applied pow1/3 0.4

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

11. Taylor expanded around -inf 64.0

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

12. Simplified 0.4

    \[\leadsto \left(\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \color{blue}{\left(\sqrt[3]{e^{0 + \log y}}\right)}\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019208 +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))