a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 3.4s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r10423 = m;
        double r10424 = 1.0;
        double r10425 = r10424 - r10423;
        double r10426 = r10423 * r10425;
        double r10427 = v;
        double r10428 = r10426 / r10427;
        double r10429 = r10428 - r10424;
        double r10430 = r10429 * r10423;
        return r10430;
}

↓

double f(double m, double v) {
        double r10431 = m;
        double r10432 = 1.0;
        double r10433 = -r10432;
        double r10434 = r10432 - r10431;
        double r10435 = r10431 * r10434;
        double r10436 = v;
        double r10437 = r10435 / r10436;
        double r10438 = r10433 + r10437;
        double r10439 = r10431 * r10438;
        return r10439;
}

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.2

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

3. Applied add-sqr-sqrt 0.6

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

4. Applied associate-*r* 0.6

   \[\leadsto \color{blue}{\left(\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \sqrt{m}\right) \cdot \sqrt{m}}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 0.7

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

7. Applied times-frac 0.7

   \[\leadsto \left(\left(\color{blue}{\frac{m}{\sqrt{v}} \cdot \frac{1 - m}{\sqrt{v}}} - 1\right) \cdot \sqrt{m}\right) \cdot \sqrt{m}\]
8. Using strategy rm

9. Applied pow1 0.7

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

10. Applied pow1 0.7

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

11. Applied pow1 0.7

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

12. Applied pow-prod-down 0.7

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

13. Applied pow-prod-down 0.7

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020059 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))