a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 19.4s

Precision: 64

\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]

Math

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

↓

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

C

double f(double m, double v) {
        double r1070588 = m;
        double r1070589 = 1.0;
        double r1070590 = r1070589 - r1070588;
        double r1070591 = r1070588 * r1070590;
        double r1070592 = v;
        double r1070593 = r1070591 / r1070592;
        double r1070594 = r1070593 - r1070589;
        double r1070595 = r1070594 * r1070588;
        return r1070595;
}

↓

double f(double m, double v) {
        double r1070596 = m;
        double r1070597 = v;
        double r1070598 = r1070597 / r1070596;
        double r1070599 = r1070596 / r1070598;
        double r1070600 = r1070599 - r1070596;
        double r1070601 = 1.0;
        double r1070602 = r1070600 * r1070601;
        double r1070603 = r1070596 * r1070596;
        double r1070604 = r1070603 / r1070598;
        double r1070605 = r1070602 - r1070604;
        return r1070605;
}

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

   \[\leadsto \color{blue}{1 \cdot \frac{{m}^{2}}{v} - \left(1 \cdot m + \frac{{m}^{3}}{v}\right)}\]

3. Simplified 0.2

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019172 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))