Average Error: 0.2 → 0.2
Time: 3.2s
Precision: binary64
\[0 < m \land 0 < v \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
↓
\[m \cdot \mathsf{fma}\left(\frac{m}{v}, 1 - m, -1\right) \]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
m \cdot \mathsf{fma}\left(\frac{m}{v}, 1 - m, -1\right)

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))
↓
(FPCore (m v) :precision binary64 (* m (fma (/ m v) (- 1.0 m) -1.0)))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}
↓
double code(double m, double v) {
	return m * fma((m / v), (1.0 - m), -1.0);
}

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. Taylor expanded in m around 0 6.6

    \[\leadsto \color{blue}{\frac{{m}^{2}}{v} - \left(\frac{{m}^{3}}{v} + m\right)} \]

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2021215 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))