Average Error: 0.1 → 0.1
Time: 1.5m
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 + \sqrt{m}\right)\right) \cdot \left(-\sqrt{m}\right) + \left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 + \sqrt{m}\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 + \sqrt{m}\right)\right) \cdot \left(-\sqrt{m}\right) + \left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot \left(1 + \sqrt{m}\right)
double f(double m, double v) {
        double r6906945 = m;
        double r6906946 = 1.0;
        double r6906947 = r6906946 - r6906945;
        double r6906948 = r6906945 * r6906947;
        double r6906949 = v;
        double r6906950 = r6906948 / r6906949;
        double r6906951 = r6906950 - r6906946;
        double r6906952 = r6906951 * r6906947;
        return r6906952;
}
↓
double f(double m, double v) {
        double r6906953 = 1.0;
        double r6906954 = m;
        double r6906955 = r6906953 - r6906954;
        double r6906956 = r6906955 * r6906954;
        double r6906957 = v;
        double r6906958 = r6906956 / r6906957;
        double r6906959 = r6906958 - r6906953;
        double r6906960 = sqrt(r6906954);
        double r6906961 = r6906953 + r6906960;
        double r6906962 = r6906959 * r6906961;
        double r6906963 = -r6906960;
        double r6906964 = r6906962 * r6906963;
        double r6906965 = r6906964 + r6906962;
        return r6906965;
}

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

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

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

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

  5. Applied difference-of-squares0.1

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

  6. Applied associate-*r*0.1

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

  8. Applied sub-neg0.1

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

  9. Applied distribute-rgt-in0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))