Average Error: 0.1 → 0.1
Time: 4.7s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \mathsf{fma}\left(-m, 1, m\right)\]
\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) \cdot \left(1 - m\right) + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \mathsf{fma}\left(-m, 1, m\right)
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) * (1.0 - m)) + ((((m * (1.0 - m)) / v) - 1.0) * fma(-m, 1.0, m)));
}

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-cube-cbrt0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied prod-diff0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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 m)) v) 1) (- 1 m)))