a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 15.2s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r379863 = m;
        double r379864 = 1.0;
        double r379865 = r379864 - r379863;
        double r379866 = r379863 * r379865;
        double r379867 = v;
        double r379868 = r379866 / r379867;
        double r379869 = r379868 - r379864;
        double r379870 = r379869 * r379863;
        return r379870;
}

↓

double f(double m, double v) {
        double r379871 = m;
        double r379872 = v;
        double r379873 = r379871 / r379872;
        double r379874 = 1.0;
        double r379875 = r379874 - r379871;
        double r379876 = -1.0;
        double r379877 = fma(r379873, r379875, r379876);
        double r379878 = r379871 * r379877;
        return r379878;
}

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. Simplified 0.2

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

3. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019152 +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))