Average Error: 0.2 → 0.2
Time: 22.2s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right) \cdot m\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right) \cdot m
double f(double m, double v) {
        double r20379 = m;
        double r20380 = 1.0;
        double r20381 = r20380 - r20379;
        double r20382 = r20379 * r20381;
        double r20383 = v;
        double r20384 = r20382 / r20383;
        double r20385 = r20384 - r20380;
        double r20386 = r20385 * r20379;
        return r20386;
}
↓
double f(double m, double v) {
        double r20387 = 1.0;
        double r20388 = m;
        double r20389 = r20387 - r20388;
        double r20390 = v;
        double r20391 = r20389 / r20390;
        double r20392 = -r20387;
        double r20393 = fma(r20391, r20388, r20392);
        double r20394 = r20393 * r20388;
        return r20394;
}

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. Using strategy rm

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

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

  4. Applied associate-*r*0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))