Average Error: 0.2 → 0.2
Time: 29.2s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\left(\left(m + \left(-m\right)\right) - \frac{m}{\frac{v}{m \cdot m}}\right) + \mathsf{fma}\left(\frac{m}{v}, m, -m\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\left(\left(m + \left(-m\right)\right) - \frac{m}{\frac{v}{m \cdot m}}\right) + \mathsf{fma}\left(\frac{m}{v}, m, -m\right)
double f(double m, double v) {
        double r927681 = m;
        double r927682 = 1.0;
        double r927683 = r927682 - r927681;
        double r927684 = r927681 * r927683;
        double r927685 = v;
        double r927686 = r927684 / r927685;
        double r927687 = r927686 - r927682;
        double r927688 = r927687 * r927681;
        return r927688;
}
↓
double f(double m, double v) {
        double r927689 = m;
        double r927690 = -r927689;
        double r927691 = r927689 + r927690;
        double r927692 = v;
        double r927693 = r927689 * r927689;
        double r927694 = r927692 / r927693;
        double r927695 = r927689 / r927694;
        double r927696 = r927691 - r927695;
        double r927697 = r927689 / r927692;
        double r927698 = fma(r927697, r927689, r927690);
        double r927699 = r927696 + r927698;
        return r927699;
}

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 6.8

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

  3. Simplified0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied associate-/r/0.2

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

  7. Applied prod-diff0.2

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

  8. Applied associate--l+0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))