a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 18.6s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r987778 = m;
        double r987779 = 1.0;
        double r987780 = r987779 - r987778;
        double r987781 = r987778 * r987780;
        double r987782 = v;
        double r987783 = r987781 / r987782;
        double r987784 = r987783 - r987779;
        double r987785 = r987784 * r987778;
        return r987785;
}

↓

double f(double m, double v) {
        double r987786 = m;
        double r987787 = v;
        double r987788 = r987786 / r987787;
        double r987789 = r987786 * r987786;
        double r987790 = fma(r987788, r987789, r987786);
        double r987791 = -r987790;
        double r987792 = fma(r987788, r987786, r987791);
        return r987792;
}

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. Taylor expanded around 0 7.2

   \[\leadsto \color{blue}{\frac{{m}^{2}}{v} - \left(m + \frac{{m}^{3}}{v}\right)}\]

3. Simplified 0.2

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019164 +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))