Average Error: 0.3 → 0.2
Time: 18.5s
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 r1813533 = x;
        double r1813534 = log(r1813533);
        double r1813535 = r1813533 * r1813534;
        return r1813535;
}
↓
double f(double x) {
        double r1813536 = x;
        double r1813537 = cbrt(r1813536);
        double r1813538 = log(r1813537);
        double r1813539 = r1813536 + r1813536;
        double r1813540 = 1.0;
        double r1813541 = r1813540 / r1813536;
        double r1813542 = -0.3333333333333333;
        double r1813543 = pow(r1813541, r1813542);
        double r1813544 = log(r1813543);
        double r1813545 = r1813536 * r1813544;
        double r1813546 = fma(r1813538, r1813539, r1813545);
        return r1813546;
}

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-rgt-in0.4

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

  6. Simplified0.4

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

  8. Applied fma-def0.3

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

  9. Taylor expanded around inf 0.2

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

  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 2019162 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  (* x (log x)))