Average Error: 0.2 → 0.2
Time: 1.4m
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \frac{m}{v} + \left(-1 - \sqrt{\frac{1}{v} \cdot \left(m \cdot m\right)} \cdot \sqrt{\frac{1}{v} \cdot \left(m \cdot m\right)}\right) \cdot m\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
m \cdot \frac{m}{v} + \left(-1 - \sqrt{\frac{1}{v} \cdot \left(m \cdot m\right)} \cdot \sqrt{\frac{1}{v} \cdot \left(m \cdot m\right)}\right) \cdot m
double f(double m, double v) {
        double r2113646 = m;
        double r2113647 = 1.0;
        double r2113648 = r2113647 - r2113646;
        double r2113649 = r2113646 * r2113648;
        double r2113650 = v;
        double r2113651 = r2113649 / r2113650;
        double r2113652 = r2113651 - r2113647;
        double r2113653 = r2113652 * r2113646;
        return r2113653;
}
↓
double f(double m, double v) {
        double r2113654 = m;
        double r2113655 = v;
        double r2113656 = r2113654 / r2113655;
        double r2113657 = r2113654 * r2113656;
        double r2113658 = -1.0;
        double r2113659 = 1.0;
        double r2113660 = r2113659 / r2113655;
        double r2113661 = r2113654 * r2113654;
        double r2113662 = r2113660 * r2113661;
        double r2113663 = sqrt(r2113662);
        double r2113664 = r2113663 * r2113663;
        double r2113665 = r2113658 - r2113664;
        double r2113666 = r2113665 * r2113654;
        double r2113667 = r2113657 + r2113666;
        return r2113667;
}

Error

Bits error versus m

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  4. Applied sub-neg0.2

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

  5. Applied distribute-lft-in0.2

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

  6. Simplified0.2

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

  8. Applied div-inv0.2

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

  9. Applied associate-*r*0.2

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

  11. Applied add-sqr-sqrt0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 +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))