Average Error: 0.2 → 0.2
Time: 13.9s
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 r422151 = m;
        double r422152 = 1.0;
        double r422153 = r422152 - r422151;
        double r422154 = r422151 * r422153;
        double r422155 = v;
        double r422156 = r422154 / r422155;
        double r422157 = r422156 - r422152;
        double r422158 = r422157 * r422151;
        return r422158;
}
↓
double f(double m, double v) {
        double r422159 = m;
        double r422160 = v;
        double r422161 = r422159 / r422160;
        double r422162 = 1.0;
        double r422163 = r422162 - r422159;
        double r422164 = -1.0;
        double r422165 = fma(r422161, r422163, r422164);
        double r422166 = r422159 * r422165;
        return r422166;
}

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. Using strategy rm

  4. Applied *-commutative0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +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))