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

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ (+ m (fma (/ m v) (- 1.0 (* m 2.0)) -1.0)) (* (/ m v) (* m m))))
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 / v), (1.0 - (m * 2.0)), -1.0)) + ((m / v) * (m * m));
}

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 *-un-lft-identity_binary640.1

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

  5. Applied add-cube-cbrt_binary640.4

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

  6. Applied unpow-prod-down_binary640.4

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

  7. Applied times-frac_binary640.4

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

  8. Simplified0.2

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

  9. Simplified0.1

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

  10. Taylor expanded in m around 0 0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(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)))