b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 12.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

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

C

double f(double m, double v) {
        double r14597 = m;
        double r14598 = 1.0;
        double r14599 = r14598 - r14597;
        double r14600 = r14597 * r14599;
        double r14601 = v;
        double r14602 = r14600 / r14601;
        double r14603 = r14602 - r14598;
        double r14604 = r14603 * r14599;
        return r14604;
}

↓

double f(double m, double v) {
        double r14605 = 1.0;
        double r14606 = m;
        double r14607 = 1.0;
        double r14608 = r14607 - r14606;
        double r14609 = r14606 * r14608;
        double r14610 = v;
        double r14611 = r14609 / r14610;
        double r14612 = sqrt(r14607);
        double r14613 = r14612 * r14612;
        double r14614 = -r14613;
        double r14615 = fma(r14605, r14611, r14614);
        double r14616 = r14615 * r14607;
        double r14617 = r14607 - r14607;
        double r14618 = r14617 * r14607;
        double r14619 = r14608 / r14610;
        double r14620 = r14619 * r14606;
        double r14621 = r14620 - r14607;
        double r14622 = r14621 * r14606;
        double r14623 = r14618 - r14622;
        double r14624 = r14616 + r14623;
        return r14624;
}

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. Simplified 0.1

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

6. Simplified 0.1

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

8. Applied add-sqr-sqrt 0.1

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

9. Applied *-un-lft-identity 0.1

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

10. Applied prod-diff 0.1

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

11. Applied distribute-rgt-in 0.1

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

12. Applied associate-+l+ 0.1

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

13. Simplified 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020042 +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)))