a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 17.2s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r19301 = m;
        double r19302 = 1.0;
        double r19303 = r19302 - r19301;
        double r19304 = r19301 * r19303;
        double r19305 = v;
        double r19306 = r19304 / r19305;
        double r19307 = r19306 - r19302;
        double r19308 = r19307 * r19301;
        return r19308;
}

↓

double f(double m, double v) {
        double r19309 = m;
        double r19310 = v;
        double r19311 = r19309 / r19310;
        double r19312 = 1.0;
        double r19313 = r19311 * r19312;
        double r19314 = r19310 / r19309;
        double r19315 = r19309 / r19314;
        double r19316 = r19313 - r19315;
        double r19317 = r19316 - r19312;
        double r19318 = r19309 * r19317;
        return r19318;
}

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. Simplified 0.2

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

3. Taylor expanded around 0 0.2

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

4. Simplified 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019194 
(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))