Average Error: 0.1 → 0.1
Time: 22.9s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) - z\right) - y\]
\left(x \cdot \log y - z\right) - y
↓
\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) - z\right) - y
double f(double x, double y, double z) {
        double r1108732 = x;
        double r1108733 = y;
        double r1108734 = log(r1108733);
        double r1108735 = r1108732 * r1108734;
        double r1108736 = z;
        double r1108737 = r1108735 - r1108736;
        double r1108738 = r1108737 - r1108733;
        return r1108738;
}
↓
double f(double x, double y, double z) {
        double r1108739 = y;
        double r1108740 = cbrt(r1108739);
        double r1108741 = log(r1108740);
        double r1108742 = x;
        double r1108743 = r1108742 + r1108742;
        double r1108744 = r1108740 * r1108740;
        double r1108745 = cbrt(r1108744);
        double r1108746 = log(r1108745);
        double r1108747 = r1108742 * r1108746;
        double r1108748 = cbrt(r1108740);
        double r1108749 = log(r1108748);
        double r1108750 = r1108749 * r1108742;
        double r1108751 = r1108747 + r1108750;
        double r1108752 = fma(r1108741, r1108743, r1108751);
        double r1108753 = z;
        double r1108754 = r1108752 - r1108753;
        double r1108755 = r1108754 - r1108739;
        return r1108755;
}

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-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

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

  8. Applied fma-def0.1

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

  10. Applied add-cube-cbrt0.1

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

  11. Applied cbrt-prod0.1

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

  12. Applied log-prod0.1

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

  13. Applied distribute-lft-in0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  (- (- (* x (log y)) z) y))