b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 3.4s

Precision: binary64

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

↓

\[\left(\left(m + \frac{m}{v}\right) - \mathsf{fma}\left(2, \frac{m \cdot m}{v}, 1\right)\right) + \frac{{m}^{3}}{v} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (+ (- (+ m (/ m v)) (fma 2.0 (/ (* m m) v) 1.0)) (/ (pow m 3.0) v)))

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 + (m / v)) - fma(2.0, ((m * m) / v), 1.0)) + (pow(m, 3.0) / v);
}

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. Taylor expanded in m around 0 0.1

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

3. Simplified 0.1

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022077 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))