b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 18.3s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r477577 = m;
        double r477578 = 1.0;
        double r477579 = r477578 - r477577;
        double r477580 = r477577 * r477579;
        double r477581 = v;
        double r477582 = r477580 / r477581;
        double r477583 = r477582 - r477578;
        double r477584 = r477583 * r477579;
        return r477584;
}

↓

double f(double m, double v) {
        double r477585 = m;
        double r477586 = v;
        double r477587 = r477585 / r477586;
        double r477588 = 1.0;
        double r477589 = r477588 - r477585;
        double r477590 = -1.0;
        double r477591 = fma(r477587, r477589, r477590);
        double r477592 = r477591 * r477589;
        return r477592;
}

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. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(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)))