Average Error: 0.2 → 0.2
Time: 11.4s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \mathsf{fma}\left(\frac{m}{v}, 1 - m, -1\right)\]
\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 r464545 = m;
        double r464546 = 1.0;
        double r464547 = r464546 - r464545;
        double r464548 = r464545 * r464547;
        double r464549 = v;
        double r464550 = r464548 / r464549;
        double r464551 = r464550 - r464546;
        double r464552 = r464551 * r464545;
        return r464552;
}
↓
double f(double m, double v) {
        double r464553 = m;
        double r464554 = v;
        double r464555 = r464553 / r464554;
        double r464556 = 1.0;
        double r464557 = r464556 - r464553;
        double r464558 = -1.0;
        double r464559 = fma(r464555, r464557, r464558);
        double r464560 = r464553 * r464559;
        return r464560;
}

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 2019156 +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))