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

Average Error: 6.8 → 0.4

Time: 22.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r56302 = x;
        double r56303 = 1.0;
        double r56304 = r56302 - r56303;
        double r56305 = y;
        double r56306 = log(r56305);
        double r56307 = r56304 * r56306;
        double r56308 = z;
        double r56309 = r56308 - r56303;
        double r56310 = r56303 - r56305;
        double r56311 = log(r56310);
        double r56312 = r56309 * r56311;
        double r56313 = r56307 + r56312;
        double r56314 = t;
        double r56315 = r56313 - r56314;
        return r56315;
}

↓

double f(double x, double y, double z, double t) {
        double r56316 = x;
        double r56317 = 1.0;
        double r56318 = r56316 - r56317;
        double r56319 = 2.0;
        double r56320 = y;
        double r56321 = cbrt(r56320);
        double r56322 = log(r56321);
        double r56323 = r56319 * r56322;
        double r56324 = r56318 * r56323;
        double r56325 = 0.3333333333333333;
        double r56326 = r56318 * r56325;
        double r56327 = log(r56320);
        double r56328 = r56326 * r56327;
        double r56329 = r56324 + r56328;
        double r56330 = z;
        double r56331 = r56330 - r56317;
        double r56332 = log(r56317);
        double r56333 = 0.5;
        double r56334 = pow(r56320, r56319);
        double r56335 = pow(r56317, r56319);
        double r56336 = r56334 / r56335;
        double r56337 = r56333 * r56336;
        double r56338 = fma(r56317, r56320, r56337);
        double r56339 = r56332 - r56338;
        double r56340 = r56331 * r56339;
        double r56341 = r56329 + r56340;
        double r56342 = t;
        double r56343 = r56341 - r56342;
        return r56343;
}

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 0.4

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

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

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

10. Applied pow1/3 0.4

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

11. Applied log-pow 0.4

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

12. Applied associate-*r* 0.4

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

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020045 +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))