Average Error: 0.1 → 0.1
Time: 3.4m
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[(\left((\left(\frac{m}{v}\right) \cdot \left(\sqrt{m}\right) + \left(\frac{m}{v}\right))_*\right) \cdot \left(1 - \sqrt{m}\right) + -1)_* \cdot \left(1 - m\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
(\left((\left(\frac{m}{v}\right) \cdot \left(\sqrt{m}\right) + \left(\frac{m}{v}\right))_*\right) \cdot \left(1 - \sqrt{m}\right) + -1)_* \cdot \left(1 - m\right)
double f(double m, double v) {
        double r9328981 = m;
        double r9328982 = 1.0;
        double r9328983 = r9328982 - r9328981;
        double r9328984 = r9328981 * r9328983;
        double r9328985 = v;
        double r9328986 = r9328984 / r9328985;
        double r9328987 = r9328986 - r9328982;
        double r9328988 = r9328987 * r9328983;
        return r9328988;
}
↓
double f(double m, double v) {
        double r9328989 = m;
        double r9328990 = v;
        double r9328991 = r9328989 / r9328990;
        double r9328992 = sqrt(r9328989);
        double r9328993 = fma(r9328991, r9328992, r9328991);
        double r9328994 = 1.0;
        double r9328995 = r9328994 - r9328992;
        double r9328996 = -1.0;
        double r9328997 = fma(r9328993, r9328995, r9328996);
        double r9328998 = r9328994 - r9328989;
        double r9328999 = r9328997 * r9328998;
        return r9328999;
}

Error

Bits error versus m

Bits error versus v

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 associate-/l*0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied add-sqr-sqrt0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied difference-of-squares0.1

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

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

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

  10. Applied times-frac0.1

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

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

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

  12. Applied times-frac0.2

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

  13. Applied prod-diff0.2

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

  14. Simplified0.1

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

  15. Simplified0.1

    \[\leadsto \left((\left((\left(\frac{m}{v}\right) \cdot \left(\sqrt{m}\right) + \left(\frac{m}{v}\right))_*\right) \cdot \left(1 - \sqrt{m}\right) + -1)_* + \color{blue}{0}\right) \cdot \left(1 - m\right)\]

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019107 +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)))