Average Error: 0.3 → 0.3
Time: 4.5s
Precision: 64
\[x \cdot \log x\]
↓
\[x \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{1}{x}\right)\right) + x \cdot \log \left(\sqrt{x}\right)\]
x \cdot \log x
↓
x \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{1}{x}\right)\right) + x \cdot \log \left(\sqrt{x}\right)
double f(double x) {
        double r29237 = x;
        double r29238 = log(r29237);
        double r29239 = r29237 * r29238;
        return r29239;
}
↓
double f(double x) {
        double r29240 = x;
        double r29241 = -0.5;
        double r29242 = 1.0;
        double r29243 = r29242 / r29240;
        double r29244 = log(r29243);
        double r29245 = r29241 * r29244;
        double r29246 = r29240 * r29245;
        double r29247 = sqrt(r29240);
        double r29248 = log(r29247);
        double r29249 = r29240 * r29248;
        double r29250 = r29246 + r29249;
        return r29250;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.3

    \[x \cdot \log x\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied log-prod0.3

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

  9. Simplified0.3

    \[\leadsto x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\sqrt{x}}\right)} + \log \left(\sqrt[3]{\sqrt{x}}\right)\right) + x \cdot \log \left(\sqrt{x}\right)\]

  10. Taylor expanded around 0 0.3

    \[\leadsto x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{x}}\right) + \log \color{blue}{\left({x}^{\frac{1}{6}}\right)}\right) + x \cdot \log \left(\sqrt{x}\right)\]

  11. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{6}}\right) \cdot x\right)} + x \cdot \log \left(\sqrt{x}\right)\]

  12. Simplified0.3

    \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{1}{x}\right)\right)} + x \cdot \log \left(\sqrt{x}\right)\]

  13. Final simplification0.3

    \[\leadsto x \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{1}{x}\right)\right) + x \cdot \log \left(\sqrt{x}\right)\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))