Average Error: 0.2 → 0.2
Time: 21.4s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right) \cdot m\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right) \cdot m
double f(double m, double v) {
        double r16965 = m;
        double r16966 = 1.0;
        double r16967 = r16966 - r16965;
        double r16968 = r16965 * r16967;
        double r16969 = v;
        double r16970 = r16968 / r16969;
        double r16971 = r16970 - r16966;
        double r16972 = r16971 * r16965;
        return r16972;
}
↓
double f(double m, double v) {
        double r16973 = 1.0;
        double r16974 = m;
        double r16975 = r16973 - r16974;
        double r16976 = v;
        double r16977 = r16974 / r16976;
        double r16978 = -r16973;
        double r16979 = fma(r16975, r16977, r16978);
        double r16980 = r16979 * r16974;
        return r16980;
}

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. Using strategy rm

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

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

  8. Applied associate-*r*0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

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))