Average Error: 0.3 → 0.2
Time: 16.9s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(x + x\right)\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right), x + x, x \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right) + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(x + x\right)\right)
double f(double x) {
        double r44946 = x;
        double r44947 = log(r44946);
        double r44948 = r44946 * r44947;
        return r44948;
}
↓
double f(double x) {
        double r44949 = x;
        double r44950 = sqrt(r44949);
        double r44951 = cbrt(r44950);
        double r44952 = r44951 * r44951;
        double r44953 = log(r44952);
        double r44954 = r44949 + r44949;
        double r44955 = cbrt(r44949);
        double r44956 = cbrt(r44955);
        double r44957 = log(r44956);
        double r44958 = r44949 * r44957;
        double r44959 = r44957 * r44954;
        double r44960 = r44958 + r44959;
        double r44961 = fma(r44953, r44954, r44960);
        return r44961;
}

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

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

  11. Applied log-prod0.3

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

  12. Applied distribute-lft-in0.2

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

  13. Simplified0.2

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

  15. Applied add-sqr-sqrt0.2

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

  16. Applied cbrt-prod0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  (* x (log x)))