Average Error: 0.2 → 0.2
Time: 21.1s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \mathsf{fma}\left(\frac{m}{v}, 1, -\mathsf{fma}\left(\frac{m}{v}, m, 1\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
m \cdot \mathsf{fma}\left(\frac{m}{v}, 1, -\mathsf{fma}\left(\frac{m}{v}, m, 1\right)\right)
double f(double m, double v) {
        double r1051088 = m;
        double r1051089 = 1.0;
        double r1051090 = r1051089 - r1051088;
        double r1051091 = r1051088 * r1051090;
        double r1051092 = v;
        double r1051093 = r1051091 / r1051092;
        double r1051094 = r1051093 - r1051089;
        double r1051095 = r1051094 * r1051088;
        return r1051095;
}
↓
double f(double m, double v) {
        double r1051096 = m;
        double r1051097 = v;
        double r1051098 = r1051096 / r1051097;
        double r1051099 = 1.0;
        double r1051100 = fma(r1051098, r1051096, r1051099);
        double r1051101 = -r1051100;
        double r1051102 = fma(r1051098, r1051099, r1051101);
        double r1051103 = r1051096 * r1051102;
        return r1051103;
}

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied associate-*r*0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))