Average Error: 0.2 → 0.2
Time: 42.4s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(\frac{m}{v}, m, -\mathsf{fma}\left(m, m \cdot \frac{m}{v}, m\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(\frac{m}{v}, m, -\mathsf{fma}\left(m, m \cdot \frac{m}{v}, m\right)\right)
double f(double m, double v) {
        double r685212 = m;
        double r685213 = 1.0;
        double r685214 = r685213 - r685212;
        double r685215 = r685212 * r685214;
        double r685216 = v;
        double r685217 = r685215 / r685216;
        double r685218 = r685217 - r685213;
        double r685219 = r685218 * r685212;
        return r685219;
}
↓
double f(double m, double v) {
        double r685220 = m;
        double r685221 = v;
        double r685222 = r685220 / r685221;
        double r685223 = r685220 * r685222;
        double r685224 = fma(r685220, r685223, r685220);
        double r685225 = -r685224;
        double r685226 = fma(r685222, r685220, r685225);
        return r685226;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{m}{\frac{v}{m}} - \mathsf{fma}\left(\frac{m}{\frac{v}{m}}, m, m\right)}\]
  3. Using strategy rm

  4. Applied associate-/r/0.2

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

  5. Applied fma-neg0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))