b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 14.9s

Precision: 64

\[0 \lt m \land 0 \lt v \land v \lt 0.25\]

Math

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

↓

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

C

double f(double m, double v) {
        double r534443 = m;
        double r534444 = 1.0;
        double r534445 = r534444 - r534443;
        double r534446 = r534443 * r534445;
        double r534447 = v;
        double r534448 = r534446 / r534447;
        double r534449 = r534448 - r534444;
        double r534450 = r534449 * r534445;
        return r534450;
}

↓

double f(double m, double v) {
        double r534451 = m;
        double r534452 = v;
        double r534453 = 1.0;
        double r534454 = r534451 * r534451;
        double r534455 = r534453 - r534454;
        double r534456 = r534452 / r534455;
        double r534457 = r534451 / r534456;
        double r534458 = r534451 + r534453;
        double r534459 = r534457 / r534458;
        double r534460 = r534459 - r534453;
        double r534461 = r534453 - r534451;
        double r534462 = r534460 * r534461;
        return r534462;
}

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. Taylor expanded around 0 0.1

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

3. Simplified 0.1

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

5. Applied *-un-lft-identity 0.1

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

6. Applied times-frac 0.1

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

7. Applied *-un-lft-identity 0.1

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

8. Applied *-un-lft-identity 0.1

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

9. Applied times-frac 0.1

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

10. Applied distribute-rgt-out-- 0.1

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

11. Simplified 0.1

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

13. Applied flip-- 0.1

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

14. Applied associate-*r/ 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019156 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))