Average Error: 0.3 → 0.3
Time: 24.2s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, x \cdot \log \left(\sqrt[3]{x}\right)\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, x \cdot \log \left(\sqrt[3]{x}\right)\right)
double f(double x) {
        double r44112 = x;
        double r44113 = log(r44112);
        double r44114 = r44112 * r44113;
        return r44114;
}
↓
double f(double x) {
        double r44115 = 2.0;
        double r44116 = x;
        double r44117 = cbrt(r44116);
        double r44118 = log(r44117);
        double r44119 = r44115 * r44118;
        double r44120 = r44116 * r44118;
        double r44121 = fma(r44119, r44116, r44120);
        return r44121;
}

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. Using strategy rm

  8. Applied fma-def0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  (* x (log x)))