Average Error: 0.2 → 0.2
Time: 17.4s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\left(\frac{m}{v} - \mathsf{fma}\left(\frac{m}{v}, m, 1\right)\right) \cdot m\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\left(\frac{m}{v} - \mathsf{fma}\left(\frac{m}{v}, m, 1\right)\right) \cdot m
double f(double m, double v) {
        double r940590 = m;
        double r940591 = 1.0;
        double r940592 = r940591 - r940590;
        double r940593 = r940590 * r940592;
        double r940594 = v;
        double r940595 = r940593 / r940594;
        double r940596 = r940595 - r940591;
        double r940597 = r940596 * r940590;
        return r940597;
}
↓
double f(double m, double v) {
        double r940598 = m;
        double r940599 = v;
        double r940600 = r940598 / r940599;
        double r940601 = 1.0;
        double r940602 = fma(r940600, r940598, r940601);
        double r940603 = r940600 - r940602;
        double r940604 = r940603 * r940598;
        return r940604;
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 +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))