Average Error: 0.1 → 0.1
Time: 4.7s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\mathsf{fma}\left(\frac{m}{\frac{v}{1 - m}}, 1, -1 \cdot 1\right) + \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot \left(-m\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\mathsf{fma}\left(\frac{m}{\frac{v}{1 - m}}, 1, -1 \cdot 1\right) + \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot \left(-m\right)
double f(double m, double v) {
        double r15889 = m;
        double r15890 = 1.0;
        double r15891 = r15890 - r15889;
        double r15892 = r15889 * r15891;
        double r15893 = v;
        double r15894 = r15892 / r15893;
        double r15895 = r15894 - r15890;
        double r15896 = r15895 * r15891;
        return r15896;
}
↓
double f(double m, double v) {
        double r15897 = m;
        double r15898 = v;
        double r15899 = 1.0;
        double r15900 = r15899 - r15897;
        double r15901 = r15898 / r15900;
        double r15902 = r15897 / r15901;
        double r15903 = r15899 * r15899;
        double r15904 = -r15903;
        double r15905 = fma(r15902, r15899, r15904);
        double r15906 = r15899 * r15897;
        double r15907 = -r15897;
        double r15908 = r15907 * r15897;
        double r15909 = r15906 + r15908;
        double r15910 = r15909 / r15898;
        double r15911 = r15910 - r15899;
        double r15912 = r15911 * r15907;
        double r15913 = r15905 + r15912;
        return r15913;
}

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 sub-neg0.1

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

  4. Applied distribute-lft-in0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  8. Applied sub-neg0.1

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

  9. Applied distribute-lft-in0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 +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)))