Average Error: 0.1 → 0.1
Time: 21.9s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\mathsf{fma}\left(1, m, m \cdot \frac{m \cdot m}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\mathsf{fma}\left(1, m, m \cdot \frac{m \cdot m}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)
double f(double m, double v) {
        double r23593 = m;
        double r23594 = 1.0;
        double r23595 = r23594 - r23593;
        double r23596 = r23593 * r23595;
        double r23597 = v;
        double r23598 = r23596 / r23597;
        double r23599 = r23598 - r23594;
        double r23600 = r23599 * r23595;
        return r23600;
}
↓
double f(double m, double v) {
        double r23601 = 1.0;
        double r23602 = m;
        double r23603 = r23601 - r23602;
        double r23604 = r23602 * r23603;
        double r23605 = v;
        double r23606 = r23604 / r23605;
        double r23607 = r23606 - r23601;
        double r23608 = r23601 * r23607;
        double r23609 = r23602 * r23602;
        double r23610 = r23609 / r23605;
        double r23611 = r23602 * r23610;
        double r23612 = fma(r23601, r23602, r23611);
        double r23613 = 2.0;
        double r23614 = pow(r23602, r23613);
        double r23615 = r23614 / r23605;
        double r23616 = r23601 * r23615;
        double r23617 = r23612 - r23616;
        double r23618 = r23608 + r23617;
        return r23618;
}

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. Simplified0.1

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

  6. Simplified0.1

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

  7. Taylor expanded around 0 0.1

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

  8. Simplified0.1

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

  10. Applied *-un-lft-identity0.1

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

  11. Applied cube-mult0.1

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

  12. Applied times-frac0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))