Average Error: 0.2 → 0.2
Time: 14.8s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \mathsf{fma}\left(\frac{m}{v}, 1 - m, -1\right)\]
\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)
double f(double m, double v) {
        double r543717 = m;
        double r543718 = 1.0;
        double r543719 = r543718 - r543717;
        double r543720 = r543717 * r543719;
        double r543721 = v;
        double r543722 = r543720 / r543721;
        double r543723 = r543722 - r543718;
        double r543724 = r543723 * r543717;
        return r543724;
}
↓
double f(double m, double v) {
        double r543725 = m;
        double r543726 = v;
        double r543727 = r543725 / r543726;
        double r543728 = 1.0;
        double r543729 = r543728 - r543725;
        double r543730 = -1.0;
        double r543731 = fma(r543727, r543729, r543730);
        double r543732 = r543725 * r543731;
        return r543732;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +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))