Average Error: 0.1 → 0.1
Time: 21.9s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\frac{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}{\sqrt{1} - \sqrt{m}} \cdot \left(\sqrt{1} - \sqrt{m}\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\frac{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}{\sqrt{1} - \sqrt{m}} \cdot \left(\sqrt{1} - \sqrt{m}\right)
double f(double m, double v) {
        double r27007 = m;
        double r27008 = 1.0;
        double r27009 = r27008 - r27007;
        double r27010 = r27007 * r27009;
        double r27011 = v;
        double r27012 = r27010 / r27011;
        double r27013 = r27012 - r27008;
        double r27014 = r27013 * r27009;
        return r27014;
}
↓
double f(double m, double v) {
        double r27015 = 1.0;
        double r27016 = m;
        double r27017 = r27015 - r27016;
        double r27018 = r27016 * r27017;
        double r27019 = v;
        double r27020 = r27018 / r27019;
        double r27021 = r27020 - r27015;
        double r27022 = r27017 * r27021;
        double r27023 = sqrt(r27015);
        double r27024 = sqrt(r27016);
        double r27025 = r27023 - r27024;
        double r27026 = r27022 / r27025;
        double r27027 = r27026 * r27025;
        return r27027;
}

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 add-sqr-sqrt0.1

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

  5. Applied difference-of-squares0.1

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

  6. Applied associate-*r*0.1

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

  8. Applied flip-+0.1

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

  9. Applied associate-*r/0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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)))