Average Error: 0.1 → 0.1
Time: 4.4s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \mathsf{fma}\left(m, 1, \frac{{m}^{3}}{v} - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)\]
\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 + \mathsf{fma}\left(m, 1, \frac{{m}^{3}}{v} - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)
double f(double m, double v) {
        double r14588 = m;
        double r14589 = 1.0;
        double r14590 = r14589 - r14588;
        double r14591 = r14588 * r14590;
        double r14592 = v;
        double r14593 = r14591 / r14592;
        double r14594 = r14593 - r14589;
        double r14595 = r14594 * r14590;
        return r14595;
}
↓
double f(double m, double v) {
        double r14596 = m;
        double r14597 = 1.0;
        double r14598 = r14597 - r14596;
        double r14599 = r14596 * r14598;
        double r14600 = v;
        double r14601 = r14599 / r14600;
        double r14602 = r14601 - r14597;
        double r14603 = r14602 * r14597;
        double r14604 = 3.0;
        double r14605 = pow(r14596, r14604);
        double r14606 = r14605 / r14600;
        double r14607 = 2.0;
        double r14608 = pow(r14596, r14607);
        double r14609 = r14608 / r14600;
        double r14610 = sqrt(r14609);
        double r14611 = r14610 * r14610;
        double r14612 = r14597 * r14611;
        double r14613 = r14606 - r14612;
        double r14614 = fma(r14596, r14597, r14613);
        double r14615 = r14603 + r14614;
        return r14615;
}

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

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

  8. Applied add-sqr-sqrt0.1

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

  9. Final simplification0.1

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

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)))