Average Error: 0.1 → 0.1
Time: 8.3s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{m} \cdot \sqrt{m}\\ t_1 := \frac{m \cdot \left(1 - m\right)}{v} - 1\\ \mathsf{fma}\left(1, 1, -t_0\right) \cdot t_1 + t_1 \cdot \left(t_0 - m\right) \end{array} \]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\begin{array}{l}
t_0 := \sqrt{m} \cdot \sqrt{m}\\
t_1 := \frac{m \cdot \left(1 - m\right)}{v} - 1\\
\mathsf{fma}\left(1, 1, -t_0\right) \cdot t_1 + t_1 \cdot \left(t_0 - m\right)
\end{array}

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* (sqrt m) (sqrt m))) (t_1 (- (/ (* m (- 1.0 m)) v) 1.0)))
   (+ (* (fma 1.0 1.0 (- t_0)) t_1) (* t_1 (- t_0 m)))))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}
↓
double code(double m, double v) {
	double t_0 = sqrt(m) * sqrt(m);
	double t_1 = ((m * (1.0 - m)) / v) - 1.0;
	return (fma(1.0, 1.0, -t_0) * t_1) + (t_1 * (t_0 - 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. Applied add-sqr-sqrt_binary640.1

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

  3. Applied *-un-lft-identity_binary640.1

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

  4. Applied prod-diff_binary640.1

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

  5. Applied distribute-rgt-in_binary640.1

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

  6. Applied fma-udef_binary640.1

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

  7. Simplified0.1

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

  8. Applied pow1_binary640.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021273 
(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)))