Average Error: 0.1 → 0.1
Time: 5.5s
Precision: binary64
\[0 < m \land 0 < v \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right) + \frac{{m}^{3}}{v}\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\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 (/ m v)) (+ 1.0 (* 2.0 (* m (/ m v))))) (/ (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 + (m / v)) - (1.0 + (2.0 * (m * (m / v))))) + (pow(m, 3.0) / v);
}

Error

Bits error versus m

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt_binary6428.3

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

  4. Simplified28.2

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

  5. Simplified28.3

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

  6. Taylor expanded around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021148 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))