Average Error: 0.1 → 0.1
Time: 4.0s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \left(\sqrt{\frac{{m}^{2}}{v}} \cdot \sqrt{\frac{{m}^{2}}{v}}\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \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 r13107 = m;
        double r13108 = 1.0;
        double r13109 = r13108 - r13107;
        double r13110 = r13107 * r13109;
        double r13111 = v;
        double r13112 = r13110 / r13111;
        double r13113 = r13112 - r13108;
        double r13114 = r13113 * r13109;
        return r13114;
}
↓
double f(double m, double v) {
        double r13115 = m;
        double r13116 = 1.0;
        double r13117 = r13116 - r13115;
        double r13118 = r13115 * r13117;
        double r13119 = v;
        double r13120 = r13118 / r13119;
        double r13121 = r13120 - r13116;
        double r13122 = r13121 * r13116;
        double r13123 = r13116 * r13115;
        double r13124 = 3.0;
        double r13125 = pow(r13115, r13124);
        double r13126 = r13125 / r13119;
        double r13127 = r13123 + r13126;
        double r13128 = 2.0;
        double r13129 = pow(r13115, r13128);
        double r13130 = r13129 / r13119;
        double r13131 = sqrt(r13130);
        double r13132 = r13131 * r13131;
        double r13133 = r13116 * r13132;
        double r13134 = r13127 - r13133;
        double r13135 = r13122 + r13134;
        return r13135;
}

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 add-sqr-sqrt0.1

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

  8. Final simplification0.1

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

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)))