Average Error: 6.7 → 0.7
Time: 9.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\sqrt{1} \cdot \sqrt{1}\right) \cdot \log y + \mathsf{fma}\left(\mathsf{fma}\left(-1, 1, 1\right), \log y, \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\sqrt{1} \cdot \sqrt{1}\right) \cdot \log y + \mathsf{fma}\left(\mathsf{fma}\left(-1, 1, 1\right), \log y, \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r60047 = x;
        double r60048 = 1.0;
        double r60049 = r60047 - r60048;
        double r60050 = y;
        double r60051 = log(r60050);
        double r60052 = r60049 * r60051;
        double r60053 = z;
        double r60054 = r60053 - r60048;
        double r60055 = r60048 - r60050;
        double r60056 = log(r60055);
        double r60057 = r60054 * r60056;
        double r60058 = r60052 + r60057;
        double r60059 = t;
        double r60060 = r60058 - r60059;
        return r60060;
}
↓
double f(double x, double y, double z, double t) {
        double r60061 = x;
        double r60062 = cbrt(r60061);
        double r60063 = r60062 * r60062;
        double r60064 = 1.0;
        double r60065 = sqrt(r60064);
        double r60066 = r60065 * r60065;
        double r60067 = -r60066;
        double r60068 = fma(r60063, r60062, r60067);
        double r60069 = y;
        double r60070 = log(r60069);
        double r60071 = r60068 * r60070;
        double r60072 = -r60064;
        double r60073 = 1.0;
        double r60074 = fma(r60072, r60073, r60064);
        double r60075 = z;
        double r60076 = r60075 - r60064;
        double r60077 = log(r60064);
        double r60078 = r60064 * r60069;
        double r60079 = 0.5;
        double r60080 = 2.0;
        double r60081 = pow(r60069, r60080);
        double r60082 = pow(r60064, r60080);
        double r60083 = r60081 / r60082;
        double r60084 = r60079 * r60083;
        double r60085 = r60078 + r60084;
        double r60086 = r60077 - r60085;
        double r60087 = r60076 * r60086;
        double r60088 = t;
        double r60089 = r60087 - r60088;
        double r60090 = fma(r60074, r60070, r60089);
        double r60091 = r60071 + r60090;
        return r60091;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.7

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

  2. Simplified6.7

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

  3. Taylor expanded around 0 0.4

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

  5. Applied fma-udef0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied add-cube-cbrt0.7

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

  9. Applied prod-diff0.7

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

  10. Applied distribute-rgt-in0.7

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

  11. Applied associate-+l+0.7

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

  12. Simplified0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2019353 +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))