Average Error: 0.1 → 0.1
Time: 4.2s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[\left(\left(m + \frac{\mathsf{fma}\left(m \cdot m, -2, m\right)}{v}\right) - 1\right) + \frac{{m}^{3}}{v} \]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\left(m + \frac{\mathsf{fma}\left(m \cdot m, -2, m\right)}{v}\right) - 1\right) + \frac{{m}^{3}}{v}

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ (- (+ m (/ (fma (* m m) -2.0 m) v)) 1.0) (/ (pow m 3.0) v)))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}
↓
double code(double m, double v) {
	return ((m + (fma((m * m), -2.0, m) / v)) - 1.0) + (pow(m, 3.0) / v);
}

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded in m around 0 0.1

    \[\leadsto \color{blue}{\left(\frac{{m}^{3}}{v} + \left(\frac{m}{v} + m\right)\right) - \left(1 + 2 \cdot \frac{{m}^{2}}{v}\right)} \]

  3. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(m + \frac{m}{v}\right) - \mathsf{fma}\left(2, \frac{m}{v} \cdot m, 1\right)\right) + \frac{{m}^{3}}{v}} \]

  4. Applied fma-udef_binary640.1

    \[\leadsto \left(\left(m + \frac{m}{v}\right) - \color{blue}{\left(2 \cdot \left(\frac{m}{v} \cdot m\right) + 1\right)}\right) + \frac{{m}^{3}}{v} \]

  5. Applied associate--r+_binary640.1

    \[\leadsto \color{blue}{\left(\left(\left(m + \frac{m}{v}\right) - 2 \cdot \left(\frac{m}{v} \cdot m\right)\right) - 1\right)} + \frac{{m}^{3}}{v} \]

  6. Simplified0.1

    \[\leadsto \left(\color{blue}{\left(m + \frac{m - 2 \cdot \left(m \cdot m\right)}{v}\right)} - 1\right) + \frac{{m}^{3}}{v} \]

  7. Taylor expanded in m around 0 0.1

    \[\leadsto \left(\left(m + \frac{\color{blue}{m - 2 \cdot {m}^{2}}}{v}\right) - 1\right) + \frac{{m}^{3}}{v} \]

  8. Simplified0.1

    \[\leadsto \left(\left(m + \frac{\color{blue}{\mathsf{fma}\left(m \cdot m, -2, m\right)}}{v}\right) - 1\right) + \frac{{m}^{3}}{v} \]

  9. Final simplification0.1

    \[\leadsto \left(\left(m + \frac{\mathsf{fma}\left(m \cdot m, -2, m\right)}{v}\right) - 1\right) + \frac{{m}^{3}}{v} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))