Average Error: 0.1 → 0.1
Time: 22.0s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\mathsf{fma}\left(-\sqrt{m}, \sqrt{m}, \sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) + \mathsf{fma}\left(1, 1, -\sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\mathsf{fma}\left(-\sqrt{m}, \sqrt{m}, \sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) + \mathsf{fma}\left(1, 1, -\sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right)
double f(double m, double v) {
        double r1222598 = m;
        double r1222599 = 1.0;
        double r1222600 = r1222599 - r1222598;
        double r1222601 = r1222598 * r1222600;
        double r1222602 = v;
        double r1222603 = r1222601 / r1222602;
        double r1222604 = r1222603 - r1222599;
        double r1222605 = r1222604 * r1222600;
        return r1222605;
}
↓
double f(double m, double v) {
        double r1222606 = m;
        double r1222607 = sqrt(r1222606);
        double r1222608 = -r1222607;
        double r1222609 = r1222607 * r1222607;
        double r1222610 = fma(r1222608, r1222607, r1222609);
        double r1222611 = v;
        double r1222612 = 1.0;
        double r1222613 = r1222612 - r1222606;
        double r1222614 = r1222611 / r1222613;
        double r1222615 = r1222606 / r1222614;
        double r1222616 = r1222615 - r1222612;
        double r1222617 = r1222610 * r1222616;
        double r1222618 = 1.0;
        double r1222619 = -r1222609;
        double r1222620 = fma(r1222618, r1222612, r1222619);
        double r1222621 = r1222620 * r1222616;
        double r1222622 = r1222617 + r1222621;
        return r1222622;
}

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. Using strategy rm

  3. Applied associate-/l*0.1

    \[\leadsto \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} - 1\right) \cdot \left(1 - m\right)\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied prod-diff0.1

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

  8. Applied distribute-lft-in0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))