Average Error: 0.1 → 0.1
Time: 5.6s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m}{\frac{v}{1 - m}} - 1\right) \cdot 1 + \left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\frac{m}{\frac{v}{1 - m}} - 1\right) \cdot 1 + \left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)
double f(double m, double v) {
        double r21511 = m;
        double r21512 = 1.0;
        double r21513 = r21512 - r21511;
        double r21514 = r21511 * r21513;
        double r21515 = v;
        double r21516 = r21514 / r21515;
        double r21517 = r21516 - r21512;
        double r21518 = r21517 * r21513;
        return r21518;
}
↓
double f(double m, double v) {
        double r21519 = m;
        double r21520 = v;
        double r21521 = 1.0;
        double r21522 = r21521 - r21519;
        double r21523 = r21520 / r21522;
        double r21524 = r21519 / r21523;
        double r21525 = r21524 - r21521;
        double r21526 = r21525 * r21521;
        double r21527 = r21521 * r21519;
        double r21528 = 3.0;
        double r21529 = pow(r21519, r21528);
        double r21530 = r21529 / r21520;
        double r21531 = r21527 + r21530;
        double r21532 = 2.0;
        double r21533 = pow(r21519, r21532);
        double r21534 = r21533 / r21520;
        double r21535 = r21521 * r21534;
        double r21536 = r21531 - r21535;
        double r21537 = r21526 + r21536;
        return r21537;
}

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. Taylor expanded around 0 0.1

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

  7. Applied associate-/l*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019318 
(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)))