a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 3.4s

Precision: binary64

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

Math

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

↓

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

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

↓

(FPCore (m v)
 :precision binary64
 (- (* 1.0 (- (* m (/ m v)) m)) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	return ((double) (((double) ((((double) (m * ((double) (1.0 - m)))) / v) - 1.0)) * m));
}

↓

double code(double m, double v) {
	return ((double) (((double) (1.0 * ((double) (((double) (m * (m / v))) - m)))) - ((double) ((m / v) * ((double) (m * m))))));
}

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 clear-num 0.2

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

4. Taylor expanded around 0 6.7

   \[\leadsto \color{blue}{1 \cdot \frac{{m}^{2}}{v} - \left(1 \cdot m + \frac{{m}^{3}}{v}\right)}\]

5. Simplified 0.2

   \[\leadsto \color{blue}{1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \frac{{m}^{3}}{v}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.2

   \[\leadsto 1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \frac{{m}^{3}}{\color{blue}{1 \cdot v}}\]

8. Applied add-cube-cbrt 0.5

   \[\leadsto 1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \frac{{\color{blue}{\left(\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right) \cdot \sqrt[3]{m}\right)}}^{3}}{1 \cdot v}\]

9. Applied unpow-prod-down 0.5

   \[\leadsto 1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \frac{\color{blue}{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{3} \cdot {\left(\sqrt[3]{m}\right)}^{3}}}{1 \cdot v}\]

10. Applied times-frac 0.5

    \[\leadsto 1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \color{blue}{\frac{{\left(\sqrt[3]{m} \cdot \sqrt[3]{m}\right)}^{3}}{1} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{3}}{v}}\]

11. Simplified 0.3

    \[\leadsto 1 \cdot \left(\frac{m}{v} \cdot m - m\right) - \color{blue}{\left(m \cdot m\right)} \cdot \frac{{\left(\sqrt[3]{m}\right)}^{3}}{v}\]

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020198 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))