Average Error: 6.9 → 0.3
Time: 22.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right) + \left(z - 1\right) \cdot \mathsf{fma}\left(-\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), 1, \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right) + \left(z - 1\right) \cdot \mathsf{fma}\left(-\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), 1, \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r139878 = x;
        double r139879 = 1.0;
        double r139880 = r139878 - r139879;
        double r139881 = y;
        double r139882 = log(r139881);
        double r139883 = r139880 * r139882;
        double r139884 = z;
        double r139885 = r139884 - r139879;
        double r139886 = r139879 - r139881;
        double r139887 = log(r139886);
        double r139888 = r139885 * r139887;
        double r139889 = r139883 + r139888;
        double r139890 = t;
        double r139891 = r139889 - r139890;
        return r139891;
}
↓
double f(double x, double y, double z, double t) {
        double r139892 = x;
        double r139893 = 1.0;
        double r139894 = r139892 - r139893;
        double r139895 = y;
        double r139896 = log(r139895);
        double r139897 = log(r139893);
        double r139898 = cbrt(r139897);
        double r139899 = r139898 * r139898;
        double r139900 = r139893 * r139895;
        double r139901 = 0.5;
        double r139902 = 2.0;
        double r139903 = pow(r139895, r139902);
        double r139904 = pow(r139893, r139902);
        double r139905 = r139903 / r139904;
        double r139906 = r139901 * r139905;
        double r139907 = r139900 + r139906;
        double r139908 = 1.0;
        double r139909 = r139907 * r139908;
        double r139910 = -r139909;
        double r139911 = fma(r139899, r139898, r139910);
        double r139912 = z;
        double r139913 = r139912 - r139893;
        double r139914 = r139911 * r139913;
        double r139915 = fma(r139894, r139896, r139914);
        double r139916 = -r139907;
        double r139917 = fma(r139916, r139908, r139909);
        double r139918 = r139913 * r139917;
        double r139919 = r139915 + r139918;
        double r139920 = t;
        double r139921 = r139919 - r139920;
        return r139921;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.9

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

  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. Using strategy rm

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

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied prod-diff0.3

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

  7. Applied distribute-lft-in0.3

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

  8. Applied associate-+r+0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020100 +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))