a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 5.9s

Precision: 64

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

Math

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

↓

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

C

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

↓

double code(double m, double v) {
	return (((1.0 / ((v / m) / (1.0 - m))) - 1.0) * 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 sub-neg 0.2

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

4. Applied distribute-lft-in 0.2

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

5. Simplified 0.2

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

6. Simplified 0.2

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

8. Applied clear-num 0.2

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

9. Simplified 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(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))