b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 4.1s

Precision: binary64

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (* (- (/ (/ (- m (pow m 3.0)) (+ m 1.0)) v) 1.0) (- 1.0 m)))

C

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

↓

double code(double m, double v) {
	return ((((m - pow(m, 3.0)) / (m + 1.0)) / v) - 1.0) * (1.0 - m);
}

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 flip--_binary64 0.1

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

4. Applied associate-*r/_binary64 0.1

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

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020219 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))