Average Error: 0.1 → 0.1
Time: 11.5s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\left(1 \cdot 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 r13725 = m;
        double r13726 = 1.0;
        double r13727 = r13726 - r13725;
        double r13728 = r13725 * r13727;
        double r13729 = v;
        double r13730 = r13728 / r13729;
        double r13731 = r13730 - r13726;
        double r13732 = r13731 * r13727;
        return r13732;
}
↓
double f(double m, double v) {
        double r13733 = 1.0;
        double r13734 = m;
        double r13735 = r13733 - r13734;
        double r13736 = r13734 * r13735;
        double r13737 = v;
        double r13738 = r13736 / r13737;
        double r13739 = r13738 - r13733;
        double r13740 = r13733 * r13739;
        double r13741 = r13733 * r13734;
        double r13742 = 3.0;
        double r13743 = pow(r13734, r13742);
        double r13744 = r13743 / r13737;
        double r13745 = r13741 + r13744;
        double r13746 = 2.0;
        double r13747 = pow(r13734, r13746);
        double r13748 = r13747 / r13737;
        double r13749 = sqrt(r13748);
        double r13750 = r13749 * r13749;
        double r13751 = r13733 * r13750;
        double r13752 = r13745 - r13751;
        double r13753 = r13740 + r13752;
        return r13753;
}

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 0.1

    \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
  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. Using strategy rm

  9. Applied add-sqr-sqrt0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(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)))