b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 3.9s

Precision: 64

\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]

Math

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]

↓

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \left({\left(\sqrt[3]{m}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{2}}{v}\right)\right)\]

C

double f(double m, double v) {
        double r11880 = m;
        double r11881 = 1.0;
        double r11882 = r11881 - r11880;
        double r11883 = r11880 * r11882;
        double r11884 = v;
        double r11885 = r11883 / r11884;
        double r11886 = r11885 - r11881;
        double r11887 = r11886 * r11882;
        return r11887;
}

↓

double f(double m, double v) {
        double r11888 = m;
        double r11889 = 1.0;
        double r11890 = r11889 - r11888;
        double r11891 = r11888 * r11890;
        double r11892 = v;
        double r11893 = r11891 / r11892;
        double r11894 = r11893 - r11889;
        double r11895 = r11894 * r11889;
        double r11896 = 2.0;
        double r11897 = pow(r11888, r11896);
        double r11898 = r11888 / r11892;
        double r11899 = r11897 * r11898;
        double r11900 = cbrt(r11888);
        double r11901 = 4.0;
        double r11902 = pow(r11900, r11901);
        double r11903 = pow(r11900, r11896);
        double r11904 = r11903 / r11892;
        double r11905 = r11902 * r11904;
        double r11906 = r11889 * r11905;
        double r11907 = r11899 - r11906;
        double r11908 = fma(r11888, r11889, r11907);
        double r11909 = r11895 + r11908;
        return r11909;
}

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.1

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
2. Using strategy rm

3. Applied sub-neg 0.1

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)}\]

4. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)}\]

5. Taylor expanded around 0 0.1

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \color{blue}{\left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)}\]

6. Simplified 0.1

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \color{blue}{\mathsf{fma}\left(m, 1, \frac{{m}^{3}}{v} - 1 \cdot \frac{{m}^{2}}{v}\right)}\]
7. Using strategy rm

8. Applied *-un-lft-identity 0.1

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \frac{{m}^{3}}{\color{blue}{1 \cdot v}} - 1 \cdot \frac{{m}^{2}}{v}\right)\]

9. Applied add-cube-cbrt 0.3

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \frac{{\color{blue}{\left(\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right) \cdot \sqrt[3]{m}\right)}}^{3}}{1 \cdot v} - 1 \cdot \frac{{m}^{2}}{v}\right)\]

10. Applied unpow-prod-down 0.4

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \frac{\color{blue}{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{3} \cdot {\left(\sqrt[3]{m}\right)}^{3}}}{1 \cdot v} - 1 \cdot \frac{{m}^{2}}{v}\right)\]

11. Applied times-frac 0.4

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \color{blue}{\frac{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{3}}{1} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{3}}{v}} - 1 \cdot \frac{{m}^{2}}{v}\right)\]

12. Simplified 0.2

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \color{blue}{{m}^{2}} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{3}}{v} - 1 \cdot \frac{{m}^{2}}{v}\right)\]

13. Simplified 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \color{blue}{\frac{m}{v}} - 1 \cdot \frac{{m}^{2}}{v}\right)\]
14. Using strategy rm

15. Applied *-un-lft-identity 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \frac{{m}^{2}}{\color{blue}{1 \cdot v}}\right)\]

16. Applied add-cube-cbrt 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right) \cdot \sqrt[3]{m}\right)}}^{2}}{1 \cdot v}\right)\]

17. Applied unpow-prod-down 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \frac{\color{blue}{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{2} \cdot {\left(\sqrt[3]{m}\right)}^{2}}}{1 \cdot v}\right)\]

18. Applied times-frac 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{2}}{v}\right)}\right)\]

19. Simplified 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \left(\color{blue}{{\left(\sqrt[3]{m}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{2}}{v}\right)\right)\]

20. Final simplification 0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, {m}^{2} \cdot \frac{m}{v} - 1 \cdot \left({\left(\sqrt[3]{m}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{2}}{v}\right)\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))