b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 5.2s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r19190 = m;
        double r19191 = 1.0;
        double r19192 = r19191 - r19190;
        double r19193 = r19190 * r19192;
        double r19194 = v;
        double r19195 = r19193 / r19194;
        double r19196 = r19195 - r19191;
        double r19197 = r19196 * r19192;
        return r19197;
}

↓

double f(double m, double v) {
        double r19198 = 1.0;
        double r19199 = m;
        double r19200 = v;
        double r19201 = r19199 / r19200;
        double r19202 = r19198 * r19201;
        double r19203 = 2.0;
        double r19204 = pow(r19199, r19203);
        double r19205 = r19204 / r19200;
        double r19206 = r19198 + r19205;
        double r19207 = r19202 - r19206;
        double r19208 = r19198 - r19199;
        double r19209 = r19207 * r19208;
        return r19209;
}

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

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019322 
(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)))