Average Error: 0.2 → 0.2
Time: 26.4s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)
double f(double m, double v) {
        double r27101 = m;
        double r27102 = 1.0;
        double r27103 = r27102 - r27101;
        double r27104 = r27101 * r27103;
        double r27105 = v;
        double r27106 = r27104 / r27105;
        double r27107 = r27106 - r27102;
        double r27108 = r27107 * r27101;
        return r27108;
}
↓
double f(double m, double v) {
        double r27109 = m;
        double r27110 = 1.0;
        double r27111 = r27110 - r27109;
        double r27112 = v;
        double r27113 = r27111 / r27112;
        double r27114 = -r27110;
        double r27115 = fma(r27109, r27113, r27114);
        double r27116 = r27109 * r27115;
        return r27116;
}

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

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

  4. Applied times-frac0.2

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

  5. Applied fma-neg0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +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))