b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 5.1s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r11334 = m;
        double r11335 = 1.0;
        double r11336 = r11335 - r11334;
        double r11337 = r11334 * r11336;
        double r11338 = v;
        double r11339 = r11337 / r11338;
        double r11340 = r11339 - r11335;
        double r11341 = r11340 * r11336;
        return r11341;
}

↓

double f(double m, double v) {
        double r11342 = 1.0;
        double r11343 = m;
        double r11344 = v;
        double r11345 = r11343 / r11344;
        double r11346 = r11342 * r11345;
        double r11347 = 2.0;
        double r11348 = pow(r11343, r11347);
        double r11349 = r11348 / r11344;
        double r11350 = r11346 - r11349;
        double r11351 = r11350 - r11342;
        double r11352 = r11342 - r11343;
        double r11353 = r11351 * r11352;
        return r11353;
}

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.1

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
2. Using strategy rm

3. Applied associate-/l* 0.1

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

4. Taylor expanded around 0 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020034 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))