Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 4.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r22637 = x;
        double r22638 = y;
        double r22639 = log(r22638);
        double r22640 = r22637 * r22639;
        double r22641 = z;
        double r22642 = r22640 - r22641;
        double r22643 = r22642 - r22638;
        return r22643;
}

↓

double f(double x, double y, double z) {
        double r22644 = x;
        double r22645 = 2.0;
        double r22646 = y;
        double r22647 = cbrt(r22646);
        double r22648 = log(r22647);
        double r22649 = r22645 * r22648;
        double r22650 = r22647 * r22647;
        double r22651 = cbrt(r22650);
        double r22652 = log(r22651);
        double r22653 = r22644 * r22652;
        double r22654 = cbrt(r22647);
        double r22655 = log(r22654);
        double r22656 = r22644 * r22655;
        double r22657 = r22653 + r22656;
        double r22658 = fma(r22644, r22649, r22657);
        double r22659 = z;
        double r22660 = r22658 - r22659;
        double r22661 = r22660 - r22646;
        return r22661;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.1

   \[\left(x \cdot \log y - z\right) - y\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - z\right) - y\]

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

   \[\leadsto \left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - z\right) - y\]
7. Using strategy rm

8. Applied fma-def 0.1

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

10. Applied add-cube-cbrt 0.1

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

11. Applied cbrt-prod 0.1

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

12. Applied log-prod 0.1

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

13. Applied distribute-lft-in 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))