b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 4.7s

Precision: 64

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

Math

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

↓

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

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 (fma((m / (v / (1.0 - m))), 1.0, -(1.0 * 1.0)) + (((((1.0 * m) + (-m * m)) / v) - 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 sub-neg 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-lft-in 0.1

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

5. Simplified 0.1

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

6. Simplified 0.1

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

8. Applied sub-neg 0.1

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

9. Applied distribute-lft-in 0.1

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

10. Simplified 0.1

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

11. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020056 +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)))