a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 19.1s

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 r864151 = m;
        double r864152 = 1.0;
        double r864153 = r864152 - r864151;
        double r864154 = r864151 * r864153;
        double r864155 = v;
        double r864156 = r864154 / r864155;
        double r864157 = r864156 - r864152;
        double r864158 = r864157 * r864151;
        return r864158;
}

↓

double f(double m, double v) {
        double r864159 = m;
        double r864160 = v;
        double r864161 = r864159 / r864160;
        double r864162 = r864159 * r864159;
        double r864163 = fma(r864161, r864162, r864159);
        double r864164 = -r864163;
        double r864165 = fma(r864161, r864159, r864164);
        return r864165;
}

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 6.7

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

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 2019168 +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))