Average Error: 0.3 → 0.2
Time: 16.9s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, x \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, x \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right)
double f(double x) {
        double r843434 = x;
        double r843435 = log(r843434);
        double r843436 = r843434 * r843435;
        return r843436;
}
↓
double f(double x) {
        double r843437 = x;
        double r843438 = cbrt(r843437);
        double r843439 = log(r843438);
        double r843440 = r843437 + r843437;
        double r843441 = 1.0;
        double r843442 = r843441 / r843437;
        double r843443 = -0.3333333333333333;
        double r843444 = pow(r843442, r843443);
        double r843445 = log(r843444);
        double r843446 = r843437 * r843445;
        double r843447 = fma(r843439, r843440, r843446);
        return r843447;
}

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

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

  8. Applied fma-def0.3

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

  9. Taylor expanded around inf 0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  (* x (log x)))