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

Average Error: 6.8 → 0.4

Time: 23.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r1992304 = x;
        double r1992305 = 1.0;
        double r1992306 = r1992304 - r1992305;
        double r1992307 = y;
        double r1992308 = log(r1992307);
        double r1992309 = r1992306 * r1992308;
        double r1992310 = z;
        double r1992311 = r1992310 - r1992305;
        double r1992312 = r1992305 - r1992307;
        double r1992313 = log(r1992312);
        double r1992314 = r1992311 * r1992313;
        double r1992315 = r1992309 + r1992314;
        double r1992316 = t;
        double r1992317 = r1992315 - r1992316;
        return r1992317;
}

↓

double f(double x, double y, double z, double t) {
        double r1992318 = 1.0;
        double r1992319 = log(r1992318);
        double r1992320 = 0.5;
        double r1992321 = y;
        double r1992322 = r1992321 / r1992318;
        double r1992323 = r1992322 * r1992322;
        double r1992324 = r1992321 * r1992318;
        double r1992325 = fma(r1992320, r1992323, r1992324);
        double r1992326 = r1992319 - r1992325;
        double r1992327 = z;
        double r1992328 = r1992327 - r1992318;
        double r1992329 = 3.0;
        double r1992330 = x;
        double r1992331 = r1992329 * r1992330;
        double r1992332 = cbrt(r1992321);
        double r1992333 = log(r1992332);
        double r1992334 = r1992331 * r1992333;
        double r1992335 = log(r1992321);
        double r1992336 = r1992318 * r1992335;
        double r1992337 = -r1992336;
        double r1992338 = r1992334 + r1992337;
        double r1992339 = fma(r1992326, r1992328, r1992338);
        double r1992340 = t;
        double r1992341 = r1992339 - r1992340;
        return r1992341;
}

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. Simplified 6.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z - 1, \log y \cdot \left(x - 1\right)\right) - t}\]

3. Taylor expanded around 0 0.3

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

4. Simplified 0.3

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

6. Applied sub-neg 0.3

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

7. Applied distribute-rgt-in 0.3

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

9. Applied add-cube-cbrt 0.3

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

10. Applied log-prod 0.4

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

11. Applied distribute-rgt-in 0.4

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

12. Simplified 0.4

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

14. Applied distribute-lft-out 0.4

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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))