Average Error: 7.2 → 0.3
Time: 23.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \frac{\frac{1}{2}}{\frac{1}{\frac{y \cdot y}{1}}}\right), z - 1, \mathsf{fma}\left(\log y, x, -\mathsf{fma}\left(1, \log y, t\right)\right)\right)\]
\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(y, 1, \frac{\frac{1}{2}}{\frac{1}{\frac{y \cdot y}{1}}}\right), z - 1, \mathsf{fma}\left(\log y, x, -\mathsf{fma}\left(1, \log y, t\right)\right)\right)
double f(double x, double y, double z, double t) {
        double r41855 = x;
        double r41856 = 1.0;
        double r41857 = r41855 - r41856;
        double r41858 = y;
        double r41859 = log(r41858);
        double r41860 = r41857 * r41859;
        double r41861 = z;
        double r41862 = r41861 - r41856;
        double r41863 = r41856 - r41858;
        double r41864 = log(r41863);
        double r41865 = r41862 * r41864;
        double r41866 = r41860 + r41865;
        double r41867 = t;
        double r41868 = r41866 - r41867;
        return r41868;
}
↓
double f(double x, double y, double z, double t) {
        double r41869 = 1.0;
        double r41870 = log(r41869);
        double r41871 = y;
        double r41872 = 0.5;
        double r41873 = r41871 * r41871;
        double r41874 = r41873 / r41869;
        double r41875 = r41869 / r41874;
        double r41876 = r41872 / r41875;
        double r41877 = fma(r41871, r41869, r41876);
        double r41878 = r41870 - r41877;
        double r41879 = z;
        double r41880 = r41879 - r41869;
        double r41881 = log(r41871);
        double r41882 = x;
        double r41883 = t;
        double r41884 = fma(r41869, r41881, r41883);
        double r41885 = -r41884;
        double r41886 = fma(r41881, r41882, r41885);
        double r41887 = fma(r41878, r41880, r41886);
        return r41887;
}

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

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

  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. Simplified0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied *-un-lft-identity0.3

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

  8. Applied distribute-lft-out--0.3

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

  9. Simplified0.3

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

  10. Taylor expanded around 0 0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +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))