b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 4.2s

Precision: 64

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

Math

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

↓

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

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 * (1.0 - m)) / v) - 1.0) * 1.0) + fma(m, 1.0, fma(pow(m, 3.0), (1.0 / v), -(1.0 * (pow(m, 2.0) / 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. Using strategy rm

3. Applied sub-neg 0.1

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

4. Applied distribute-lft-in 0.1

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

5. Taylor expanded around 0 0.1

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

6. Simplified 0.1

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

8. Applied div-inv 0.1

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

9. Applied fma-neg 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(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 m)) v) 1) (- 1 m)))