Average Error: 0.3 → 0.2
Time: 5.6s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), \left(x \cdot \frac{1}{3}\right) \cdot \log x\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), \left(x \cdot \frac{1}{3}\right) \cdot \log x\right)
double f(double x) {
        double r31615 = x;
        double r31616 = log(r31615);
        double r31617 = r31615 * r31616;
        return r31617;
}
↓
double f(double x) {
        double r31618 = x;
        double r31619 = 2.0;
        double r31620 = cbrt(r31618);
        double r31621 = log(r31620);
        double r31622 = r31619 * r31621;
        double r31623 = 0.3333333333333333;
        double r31624 = r31618 * r31623;
        double r31625 = log(r31618);
        double r31626 = r31624 * r31625;
        double r31627 = fma(r31618, r31622, r31626);
        return r31627;
}

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. Using strategy rm

  10. Applied pow1/30.2

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

  11. Applied log-pow0.2

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

  12. Applied associate-*r*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))