Average Error: 0.3 → 0.4
Time: 4.6s
Precision: 64
\[x \cdot \log x\]
↓
\[x \cdot \log \left(\sqrt{x}\right) + 3 \cdot \left(x \cdot \log \left({x}^{\frac{1}{6}}\right)\right)\]
x \cdot \log x
↓
x \cdot \log \left(\sqrt{x}\right) + 3 \cdot \left(x \cdot \log \left({x}^{\frac{1}{6}}\right)\right)
double code(double x) {
	return ((double) (x * ((double) log(x))));
}
↓
double code(double x) {
	return ((double) (((double) (x * ((double) log(((double) sqrt(x)))))) + ((double) (3.0 * ((double) (x * ((double) log(((double) pow(x, 0.16666666666666666))))))))));
}

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

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

  8. Applied log-prod0.3

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

  9. Applied distribute-lft-in0.3

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

  10. Simplified0.3

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

  11. Taylor expanded around 0 0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020126 
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))