a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 20.0s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r732674 = m;
        double r732675 = 1.0;
        double r732676 = r732675 - r732674;
        double r732677 = r732674 * r732676;
        double r732678 = v;
        double r732679 = r732677 / r732678;
        double r732680 = r732679 - r732675;
        double r732681 = r732680 * r732674;
        return r732681;
}

↓

double f(double m, double v) {
        double r732682 = m;
        double r732683 = r732682 * r732682;
        double r732684 = r732682 - r732683;
        double r732685 = v;
        double r732686 = r732684 / r732685;
        double r732687 = 1.0;
        double r732688 = r732686 - r732687;
        double r732689 = r732682 * r732688;
        return r732689;
}

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. Taylor expanded around inf 0.2

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

3. Simplified 0.2

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

4. Taylor expanded around 0 0.2

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019134 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))