Average Error: 0.3 → 0.2
Time: 14.3s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \log \left({\left({x}^{\frac{2}{3}}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \log \left({\left({x}^{\frac{2}{3}}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)
double f(double x) {
        double r39545 = x;
        double r39546 = log(r39545);
        double r39547 = r39545 * r39546;
        return r39547;
}
↓
double f(double x) {
        double r39548 = 2.0;
        double r39549 = x;
        double r39550 = cbrt(r39549);
        double r39551 = log(r39550);
        double r39552 = r39548 * r39551;
        double r39553 = 0.6666666666666666;
        double r39554 = pow(r39549, r39553);
        double r39555 = 0.3333333333333333;
        double r39556 = pow(r39554, r39555);
        double r39557 = cbrt(r39550);
        double r39558 = r39556 * r39557;
        double r39559 = log(r39558);
        double r39560 = r39559 * r39549;
        double r39561 = fma(r39552, r39549, r39560);
        return r39561;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.3

    \[x \cdot \log x\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

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

  4. Applied log-prod0.4

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

  5. Applied distribute-lft-in0.4

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

  6. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot x} + x \cdot \log \left(\sqrt[3]{x}\right)\]

  7. Simplified0.4

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

  9. Applied fma-def0.3

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

  11. Applied add-cube-cbrt0.3

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

  12. Applied cbrt-prod0.3

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

  13. Simplified0.2

    \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \log \left(\color{blue}{\sqrt[3]{{x}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)\]
  14. Using strategy rm

  15. Applied pow1/30.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))