a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 29.1s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r661581 = m;
        double r661582 = 1.0;
        double r661583 = r661582 - r661581;
        double r661584 = r661581 * r661583;
        double r661585 = v;
        double r661586 = r661584 / r661585;
        double r661587 = r661586 - r661582;
        double r661588 = r661587 * r661581;
        return r661588;
}

↓

double f(double m, double v) {
        double r661589 = m;
        double r661590 = v;
        double r661591 = r661589 / r661590;
        double r661592 = -r661589;
        double r661593 = fma(r661591, r661589, r661592);
        double r661594 = r661589 * r661589;
        double r661595 = r661591 * r661594;
        double r661596 = r661593 - r661595;
        return r661596;
}

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 inf 7.1

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

3. Simplified 0.2

   \[\leadsto \color{blue}{\left(\frac{m}{v} \cdot m - m\right) - \frac{m}{v} \cdot \left(m \cdot m\right)}\]
4. Using strategy rm

5. Applied fma-neg 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019141 +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))