b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 17.4s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r21822 = m;
        double r21823 = 1.0;
        double r21824 = r21823 - r21822;
        double r21825 = r21822 * r21824;
        double r21826 = v;
        double r21827 = r21825 / r21826;
        double r21828 = r21827 - r21823;
        double r21829 = r21828 * r21824;
        return r21829;
}

↓

double f(double m, double v) {
        double r21830 = m;
        double r21831 = 1.0;
        double r21832 = r21831 - r21830;
        double r21833 = r21830 * r21832;
        double r21834 = v;
        double r21835 = r21833 / r21834;
        double r21836 = r21835 - r21831;
        double r21837 = r21836 * r21832;
        return r21837;
}

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 clear-num 0.2

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

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. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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)))