Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2

Average Error: 0.3 → 0.2

Time: 13.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x \cdot \log x\]

↓

\[\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot x + \log \left(\sqrt[3]{\sqrt[3]{x}}\right) \cdot x\right)\]

C

double f(double x) {
        double r33432 = x;
        double r33433 = log(r33432);
        double r33434 = r33432 * r33433;
        return r33434;
}

↓

double f(double x) {
        double r33435 = 2.0;
        double r33436 = x;
        double r33437 = cbrt(r33436);
        double r33438 = log(r33437);
        double r33439 = r33435 * r33438;
        double r33440 = cbrt(r33437);
        double r33441 = log(r33440);
        double r33442 = r33435 * r33441;
        double r33443 = r33442 * r33436;
        double r33444 = r33441 * r33436;
        double r33445 = r33443 + r33444;
        double r33446 = fma(r33439, r33436, r33445);
        return r33446;
}

Error

Bits error versus x

Derivation

1. Initial program 0.3

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

3. Applied add-cube-cbrt 0.3

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

4. Applied log-prod 0.4

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

5. Applied distribute-lft-in 0.4

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

6. Simplified 0.4

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

8. Applied fma-def 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \log \left(\sqrt[3]{x}\right), x, x \cdot \log \left(\sqrt[3]{x}\right)\right)}\]
9. Using strategy rm

10. Applied add-cube-cbrt 0.3

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

11. Applied log-prod 0.2

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

12. Applied distribute-lft-in 0.2

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

13. Simplified 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))