a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 21.6s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r634402 = m;
        double r634403 = 1.0;
        double r634404 = r634403 - r634402;
        double r634405 = r634402 * r634404;
        double r634406 = v;
        double r634407 = r634405 / r634406;
        double r634408 = r634407 - r634403;
        double r634409 = r634408 * r634402;
        return r634409;
}

↓

double f(double m, double v) {
        double r634410 = m;
        double r634411 = v;
        double r634412 = r634411 / r634410;
        double r634413 = r634410 / r634412;
        double r634414 = r634410 * r634410;
        double r634415 = r634414 / r634412;
        double r634416 = r634415 + r634410;
        double r634417 = r634413 - r634416;
        return r634417;
}

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 associate-/l* 0.2

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

4. Taylor expanded around 0 7.2

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

5. Simplified 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019135 +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))