Average Error: 0.3 → 0.2
Time: 17.9s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right)
double f(double x) {
        double r43445 = x;
        double r43446 = log(r43445);
        double r43447 = r43445 * r43446;
        return r43447;
}
↓
double f(double x) {
        double r43448 = 2.0;
        double r43449 = x;
        double r43450 = cbrt(r43449);
        double r43451 = log(r43450);
        double r43452 = r43448 * r43451;
        double r43453 = 1.0;
        double r43454 = r43453 / r43449;
        double r43455 = -0.3333333333333333;
        double r43456 = pow(r43454, r43455);
        double r43457 = log(r43456);
        double r43458 = r43457 * r43449;
        double r43459 = fma(r43452, r43449, r43458);
        return r43459;
}

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. Taylor expanded around inf 0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))