b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 57.2s

Precision: binary64

Cost: 1344

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

Math

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

↓

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

FPCore

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

↓

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

Error

Bits error versus m

Bits error versus v

Alternative 1
Accuracy	0.1
Cost	1344

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

Alternative 2
Accuracy	0.1
Cost	832

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

Alternative 3
Accuracy	0.8
Cost	1472

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

Alternative 4
Accuracy	64.0
Cost	2496

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

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_binary64 0.1

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

4. Applied distribute-rgt-in_binary64 0.1

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

5. Simplified 0.1

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

6. Simplified 0.1

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

8. Applied *-un-lft-identity_binary64 0.1

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

9. Applied associate-/r*_binary64 0.1

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

10. Simplified 0.1

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

12. Applied sub-neg_binary64 0.1

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

13. Applied distribute-rgt-in_binary64 0.1

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

14. Simplified 0.1

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

15. Simplified 0.1

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

17. Applied *-un-lft-identity_binary64 0.1

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

18. Applied distribute-rgt-out--_binary64 0.1

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

19. Applied associate-/l*_binary64 0.1

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

20. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020322 
(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)))