Average Error: 0.1 → 0.1
Time: 11.4s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\left(\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), x, \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) - z\right) - y\]
\left(x \cdot \log y - z\right) - y
↓
\left(\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{y}\right), x, \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) - z\right) - y
double f(double x, double y, double z) {
        double r28658 = x;
        double r28659 = y;
        double r28660 = log(r28659);
        double r28661 = r28658 * r28660;
        double r28662 = z;
        double r28663 = r28661 - r28662;
        double r28664 = r28663 - r28659;
        return r28664;
}
↓
double f(double x, double y, double z) {
        double r28665 = 2.0;
        double r28666 = y;
        double r28667 = cbrt(r28666);
        double r28668 = log(r28667);
        double r28669 = r28665 * r28668;
        double r28670 = x;
        double r28671 = cbrt(r28667);
        double r28672 = log(r28671);
        double r28673 = r28665 * r28672;
        double r28674 = r28673 * r28670;
        double r28675 = r28670 * r28672;
        double r28676 = r28674 + r28675;
        double r28677 = fma(r28669, r28670, r28676);
        double r28678 = z;
        double r28679 = r28677 - r28678;
        double r28680 = r28679 - r28666;
        return r28680;
}

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

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

  8. Applied fma-def0.1

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

  10. Applied add-cube-cbrt0.1

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

  11. Applied log-prod0.1

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

  12. Applied distribute-lft-in0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

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))