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

Average Error: 0.3 → 0.3

Time: 14.6s

Precision: 64

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

Math

\[x \cdot \log x\]

↓

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

C

double f(double x) {
        double r36231 = x;
        double r36232 = log(r36231);
        double r36233 = r36231 * r36232;
        return r36233;
}

↓

double f(double x) {
        double r36234 = 2.0;
        double r36235 = x;
        double r36236 = cbrt(r36235);
        double r36237 = log(r36236);
        double r36238 = r36234 * r36237;
        double r36239 = sqrt(r36235);
        double r36240 = cbrt(r36236);
        double r36241 = log(r36240);
        double r36242 = r36234 * r36241;
        double r36243 = r36239 * r36242;
        double r36244 = r36239 * r36241;
        double r36245 = r36243 + r36244;
        double r36246 = r36239 * r36245;
        double r36247 = fma(r36238, r36235, r36246);
        return r36247;
}

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-sqr-sqrt 0.3

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

11. Applied associate-*l* 0.3

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

13. Applied add-cube-cbrt 0.3

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

14. Applied log-prod 0.3

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

15. Applied distribute-lft-in 0.3

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))