Average Error: 0.3 → 0.3
Time: 5.5s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), x \cdot \log \left(\sqrt[3]{x}\right)\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), x \cdot \log \left(\sqrt[3]{x}\right)\right)
double f(double x) {
        double r31472 = x;
        double r31473 = log(r31472);
        double r31474 = r31472 * r31473;
        return r31474;
}
↓
double f(double x) {
        double r31475 = x;
        double r31476 = 2.0;
        double r31477 = cbrt(r31475);
        double r31478 = log(r31477);
        double r31479 = r31476 * r31478;
        double r31480 = r31475 * r31478;
        double r31481 = fma(r31475, r31479, r31480);
        return r31481;
}

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

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

  8. Applied fma-def0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))