Average Error: 0.2 → 0.2
Time: 4.6s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot m\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot m
double f(double m, double v) {
        double r11100 = m;
        double r11101 = 1.0;
        double r11102 = r11101 - r11100;
        double r11103 = r11100 * r11102;
        double r11104 = v;
        double r11105 = r11103 / r11104;
        double r11106 = r11105 - r11101;
        double r11107 = r11106 * r11100;
        return r11107;
}
↓
double f(double m, double v) {
        double r11108 = m;
        double r11109 = 1.0;
        double r11110 = r11108 / r11109;
        double r11111 = 1.0;
        double r11112 = r11111 - r11108;
        double r11113 = v;
        double r11114 = r11112 / r11113;
        double r11115 = -r11111;
        double r11116 = fma(r11110, r11114, r11115);
        double r11117 = r11116 * r11108;
        return r11117;
}

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

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

Reproduce

herbie shell --seed 2020083 +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))