a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 50.7s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r1967739 = m;
        double r1967740 = 1.0;
        double r1967741 = r1967740 - r1967739;
        double r1967742 = r1967739 * r1967741;
        double r1967743 = v;
        double r1967744 = r1967742 / r1967743;
        double r1967745 = r1967744 - r1967740;
        double r1967746 = r1967745 * r1967739;
        return r1967746;
}

↓

double f(double m, double v) {
        double r1967747 = m;
        double r1967748 = v;
        double r1967749 = r1967747 / r1967748;
        double r1967750 = -r1967747;
        double r1967751 = fma(r1967749, r1967750, r1967749);
        double r1967752 = 1.0;
        double r1967753 = r1967751 - r1967752;
        double r1967754 = r1967747 * r1967753;
        return r1967754;
}

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 0 0.2

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

3. Simplified 0.2

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(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))