a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 32.5s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r1119023 = m;
        double r1119024 = 1.0;
        double r1119025 = r1119024 - r1119023;
        double r1119026 = r1119023 * r1119025;
        double r1119027 = v;
        double r1119028 = r1119026 / r1119027;
        double r1119029 = r1119028 - r1119024;
        double r1119030 = r1119029 * r1119023;
        return r1119030;
}

↓

double f(double m, double v) {
        double r1119031 = m;
        double r1119032 = v;
        double r1119033 = r1119032 / r1119031;
        double r1119034 = r1119031 / r1119033;
        double r1119035 = r1119034 - r1119031;
        double r1119036 = r1119031 * r1119031;
        double r1119037 = r1119036 * r1119031;
        double r1119038 = r1119037 / r1119032;
        double r1119039 = r1119035 - r1119038;
        return r1119039;
}

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. Using strategy rm

3. Applied div-inv 0.2

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

4. Taylor expanded around 0 6.6

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019158 +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))