Average Error: 0.3 → 0.2
Time: 19.7s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(\log \left(\sqrt[3]{x}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)
double f(double x) {
        double r1592248 = x;
        double r1592249 = log(r1592248);
        double r1592250 = r1592248 * r1592249;
        return r1592250;
}
↓
double f(double x) {
        double r1592251 = x;
        double r1592252 = cbrt(r1592251);
        double r1592253 = log(r1592252);
        double r1592254 = r1592251 + r1592251;
        double r1592255 = r1592252 * r1592252;
        double r1592256 = cbrt(r1592255);
        double r1592257 = log(r1592256);
        double r1592258 = r1592251 * r1592257;
        double r1592259 = cbrt(r1592252);
        double r1592260 = log(r1592259);
        double r1592261 = r1592260 * r1592251;
        double r1592262 = r1592258 + r1592261;
        double r1592263 = fma(r1592253, r1592254, r1592262);
        return r1592263;
}

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. Using strategy rm

  10. Applied add-cube-cbrt0.3

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

  11. Applied cbrt-prod0.3

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

  12. Applied log-prod0.2

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

  13. Applied distribute-lft-in0.2

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

  14. Final simplification0.2

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

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)))