b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 29.1s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r987638 = m;
        double r987639 = 1.0;
        double r987640 = r987639 - r987638;
        double r987641 = r987638 * r987640;
        double r987642 = v;
        double r987643 = r987641 / r987642;
        double r987644 = r987643 - r987639;
        double r987645 = r987644 * r987640;
        return r987645;
}

↓

double f(double m, double v) {
        double r987646 = 1.0;
        double r987647 = m;
        double r987648 = r987646 - r987647;
        double r987649 = v;
        double r987650 = r987647 / r987649;
        double r987651 = r987647 * r987650;
        double r987652 = r987650 - r987651;
        double r987653 = r987652 - r987646;
        double r987654 = r987648 * r987653;
        return r987654;
}

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 inf 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

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019144 
(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)))