Average Error: 0.1 → 0.1
Time: 22.5s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{{y}^{\frac{2}{3}}}\right) \cdot x\right) - \left(z + y\right)\]
\left(x \cdot \log y - z\right) - y
↓
\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{{y}^{\frac{2}{3}}}\right) \cdot x\right) - \left(z + y\right)
double f(double x, double y, double z) {
        double r30507 = x;
        double r30508 = y;
        double r30509 = log(r30508);
        double r30510 = r30507 * r30509;
        double r30511 = z;
        double r30512 = r30510 - r30511;
        double r30513 = r30512 - r30508;
        return r30513;
}
↓
double f(double x, double y, double z) {
        double r30514 = y;
        double r30515 = cbrt(r30514);
        double r30516 = log(r30515);
        double r30517 = x;
        double r30518 = r30517 + r30517;
        double r30519 = cbrt(r30515);
        double r30520 = log(r30519);
        double r30521 = r30517 * r30520;
        double r30522 = 0.6666666666666666;
        double r30523 = pow(r30514, r30522);
        double r30524 = cbrt(r30523);
        double r30525 = log(r30524);
        double r30526 = r30525 * r30517;
        double r30527 = r30521 + r30526;
        double r30528 = fma(r30516, r30518, r30527);
        double r30529 = z;
        double r30530 = r30529 + r30514;
        double r30531 = r30528 - r30530;
        return r30531;
}

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. Simplified0.1

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied distribute-lft-in0.1

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

  7. Simplified0.1

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

  9. Applied fma-def0.1

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

  11. Applied add-cube-cbrt0.1

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

  12. Applied cbrt-prod0.1

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

  13. Applied log-prod0.1

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

  14. Applied distribute-lft-in0.1

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

  15. Simplified0.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  (- (- (* x (log y)) z) y))