Average Error: 0.1 → 0.1
Time: 5.0s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\left(m - \frac{m \cdot m}{v}\right) + \frac{{m}^{3}}{v}\right) \]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\left(m - \frac{m \cdot m}{v}\right) + \frac{{m}^{3}}{v}\right)

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ (- (/ (* m (- 1.0 m)) v) 1.0) (+ (- m (/ (* 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 * (1.0 - m)) / v) - 1.0) + ((m - ((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. Applied add-cube-cbrt_binary640.2

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

  3. Applied cancel-sign-sub-inv_binary640.2

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

  4. Applied distribute-rgt-in_binary640.2

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

  5. Taylor expanded in m around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2021280 
(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)))