Average Error: 0.1 → 0.1
Time: 19.4s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\left(1 \cdot m + m \cdot \frac{m \cdot m}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)\]
\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(\left(1 \cdot m + m \cdot \frac{m \cdot m}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)
double f(double m, double v) {
        double r27866 = m;
        double r27867 = 1.0;
        double r27868 = r27867 - r27866;
        double r27869 = r27866 * r27868;
        double r27870 = v;
        double r27871 = r27869 / r27870;
        double r27872 = r27871 - r27867;
        double r27873 = r27872 * r27868;
        return r27873;
}
↓
double f(double m, double v) {
        double r27874 = 1.0;
        double r27875 = m;
        double r27876 = r27874 - r27875;
        double r27877 = r27875 * r27876;
        double r27878 = v;
        double r27879 = r27877 / r27878;
        double r27880 = r27879 - r27874;
        double r27881 = r27874 * r27880;
        double r27882 = r27874 * r27875;
        double r27883 = r27875 * r27875;
        double r27884 = r27883 / r27878;
        double r27885 = r27875 * r27884;
        double r27886 = r27882 + r27885;
        double r27887 = 2.0;
        double r27888 = pow(r27875, r27887);
        double r27889 = r27888 / r27878;
        double r27890 = r27874 * r27889;
        double r27891 = r27886 - r27890;
        double r27892 = r27881 + r27891;
        return r27892;
}

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 add-sqr-sqrt0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied difference-of-squares0.1

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

  6. Applied associate-*r*0.1

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

  8. Applied sub-neg0.1

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

  9. Applied distribute-lft-in0.1

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

  10. Simplified0.1

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

  11. Simplified0.1

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

  12. Taylor expanded around 0 0.1

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

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

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

  15. Applied cube-mult0.1

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

  16. Applied times-frac0.1

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

  17. Simplified0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(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)))