Average Error: 0.3 → 0.3
Time: 18.3s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, \frac{1}{3} \cdot \left(\left(\log x \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, \frac{1}{3} \cdot \left(\left(\log x \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)\right)
double f(double x) {
        double r1104127 = x;
        double r1104128 = log(r1104127);
        double r1104129 = r1104127 * r1104128;
        return r1104129;
}
↓
double f(double x) {
        double r1104130 = x;
        double r1104131 = cbrt(r1104130);
        double r1104132 = log(r1104131);
        double r1104133 = r1104130 + r1104130;
        double r1104134 = 0.3333333333333333;
        double r1104135 = log(r1104130);
        double r1104136 = sqrt(r1104130);
        double r1104137 = r1104135 * r1104136;
        double r1104138 = r1104137 * r1104136;
        double r1104139 = r1104134 * r1104138;
        double r1104140 = fma(r1104132, r1104133, r1104139);
        return r1104140;
}

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 pow1/30.3

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

  9. Applied log-pow0.3

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

  10. Applied associate-*l*0.3

    \[\leadsto \log \left(\sqrt[3]{x}\right) \cdot \left(x + x\right) + \color{blue}{\frac{1}{3} \cdot \left(\log x \cdot x\right)}\]
  11. Using strategy rm

  12. Applied fma-def0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, \frac{1}{3} \cdot \left(\log x \cdot x\right)\right)}\]
  13. Using strategy rm

  14. Applied add-sqr-sqrt0.3

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

  15. Applied associate-*r*0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  (* x (log x)))