Average Error: 0.3 → 0.2
Time: 6.2s
Precision: 64
\[x \cdot \log x\]
↓
\[\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), {\left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right)}^{1}\right)\]
x \cdot \log x
↓
\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), {\left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right)}^{1}\right)
double code(double x) {
	return (x * log(x));
}
↓
double code(double x) {
	return fma(x, (2.0 * log(cbrt(x))), pow((x * log(pow(x, 0.3333333333333333))), 1.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + x \cdot \log \left(\sqrt[3]{x}\right)\]
  7. Using strategy rm

  8. Applied pow10.4

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

  9. Applied pow10.4

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

  10. Applied pow-prod-down0.4

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

  11. Simplified0.3

    \[\leadsto x \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + {\color{blue}{\left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right)}}^{1}\]
  12. Using strategy rm

  13. Applied fma-def0.2

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

  14. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{x}\right), {\left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right)}^{1}\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))