b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 14.4s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r27599 = m;
        double r27600 = 1.0;
        double r27601 = r27600 - r27599;
        double r27602 = r27599 * r27601;
        double r27603 = v;
        double r27604 = r27602 / r27603;
        double r27605 = r27604 - r27600;
        double r27606 = r27605 * r27601;
        return r27606;
}

↓

double f(double m, double v) {
        double r27607 = m;
        double r27608 = v;
        double r27609 = 1.0;
        double r27610 = r27609 - r27607;
        double r27611 = r27608 / r27610;
        double r27612 = r27607 / r27611;
        double r27613 = r27612 - r27609;
        double r27614 = r27613 * r27610;
        return r27614;
}

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 *-un-lft-identity 0.1

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

4. Applied associate-*r* 0.1

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

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019195 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))