a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 6.0s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r22080 = m;
        double r22081 = 1.0;
        double r22082 = r22081 - r22080;
        double r22083 = r22080 * r22082;
        double r22084 = v;
        double r22085 = r22083 / r22084;
        double r22086 = r22085 - r22081;
        double r22087 = r22086 * r22080;
        return r22087;
}

↓

double f(double m, double v) {
        double r22088 = 1.0;
        double r22089 = m;
        double r22090 = v;
        double r22091 = r22089 / r22090;
        double r22092 = r22088 * r22091;
        double r22093 = 2.0;
        double r22094 = pow(r22089, r22093);
        double r22095 = r22094 / r22090;
        double r22096 = r22092 - r22095;
        double r22097 = r22096 - r22088;
        double r22098 = r22097 * r22089;
        return r22098;
}

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 add-sqr-sqrt 0.4

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

4. Applied times-frac 0.4

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

5. Taylor expanded around 0 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019318 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))