Average Error: 7.2 → 0.9
Time: 1.6m
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(z - 1, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y} \cdot \left(x - 1\right)\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(z - 1, \log 1 - \mathsf{fma}\left(1, y, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), \left(\sqrt[3]{\log y} \cdot \left(x - 1\right)\right) \cdot \left(\sqrt[3]{\log y} \cdot \sqrt[3]{\log y}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2469422 = x;
        double r2469423 = 1.0;
        double r2469424 = r2469422 - r2469423;
        double r2469425 = y;
        double r2469426 = log(r2469425);
        double r2469427 = r2469424 * r2469426;
        double r2469428 = z;
        double r2469429 = r2469428 - r2469423;
        double r2469430 = r2469423 - r2469425;
        double r2469431 = log(r2469430);
        double r2469432 = r2469429 * r2469431;
        double r2469433 = r2469427 + r2469432;
        double r2469434 = t;
        double r2469435 = r2469433 - r2469434;
        return r2469435;
}
↓
double f(double x, double y, double z, double t) {
        double r2469436 = z;
        double r2469437 = 1.0;
        double r2469438 = r2469436 - r2469437;
        double r2469439 = log(r2469437);
        double r2469440 = y;
        double r2469441 = 0.5;
        double r2469442 = r2469437 / r2469440;
        double r2469443 = r2469442 * r2469442;
        double r2469444 = r2469441 / r2469443;
        double r2469445 = fma(r2469437, r2469440, r2469444);
        double r2469446 = r2469439 - r2469445;
        double r2469447 = log(r2469440);
        double r2469448 = cbrt(r2469447);
        double r2469449 = x;
        double r2469450 = r2469449 - r2469437;
        double r2469451 = r2469448 * r2469450;
        double r2469452 = r2469448 * r2469448;
        double r2469453 = r2469451 * r2469452;
        double r2469454 = fma(r2469438, r2469446, r2469453);
        double r2469455 = t;
        double r2469456 = r2469454 - r2469455;
        return r2469456;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.2

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

  2. Simplified7.2

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt0.9

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

  7. Applied associate-*l*0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2019200 +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))