Average Error: 0.1 → 0.1
Time: 10.7s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \left(\mathsf{fma}\left(1, m, \frac{{m}^{3}}{v}\right) - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)\]
\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 + \left(\mathsf{fma}\left(1, m, \frac{{m}^{3}}{v}\right) - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)
double f(double m, double v) {
        double r14604 = m;
        double r14605 = 1.0;
        double r14606 = r14605 - r14604;
        double r14607 = r14604 * r14606;
        double r14608 = v;
        double r14609 = r14607 / r14608;
        double r14610 = r14609 - r14605;
        double r14611 = r14610 * r14606;
        return r14611;
}
↓
double f(double m, double v) {
        double r14612 = m;
        double r14613 = 1.0;
        double r14614 = r14613 - r14612;
        double r14615 = r14612 * r14614;
        double r14616 = v;
        double r14617 = r14615 / r14616;
        double r14618 = r14617 - r14613;
        double r14619 = r14618 * r14613;
        double r14620 = 3.0;
        double r14621 = pow(r14612, r14620);
        double r14622 = r14621 / r14616;
        double r14623 = fma(r14613, r14612, r14622);
        double r14624 = 2.0;
        double r14625 = pow(r14612, r14624);
        double r14626 = r14625 / r14616;
        double r14627 = sqrt(r14626);
        double r14628 = r14627 * r14627;
        double r14629 = r14613 * r14628;
        double r14630 = r14623 - r14629;
        double r14631 = r14619 + r14630;
        return r14631;
}

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-neg0.1

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

  4. Applied distribute-lft-in0.1

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

  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. Simplified0.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))