Average Error: 0.2 → 0.2
Time: 4.7s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(m, \frac{m}{v} \cdot \frac{1 \cdot 1 - m \cdot m}{1 + m}, -1 \cdot m\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(m, \frac{m}{v} \cdot \frac{1 \cdot 1 - m \cdot m}{1 + m}, -1 \cdot m\right)
double f(double m, double v) {
        double r12658 = m;
        double r12659 = 1.0;
        double r12660 = r12659 - r12658;
        double r12661 = r12658 * r12660;
        double r12662 = v;
        double r12663 = r12661 / r12662;
        double r12664 = r12663 - r12659;
        double r12665 = r12664 * r12658;
        return r12665;
}
↓
double f(double m, double v) {
        double r12666 = m;
        double r12667 = v;
        double r12668 = r12666 / r12667;
        double r12669 = 1.0;
        double r12670 = r12669 * r12669;
        double r12671 = r12666 * r12666;
        double r12672 = r12670 - r12671;
        double r12673 = r12669 + r12666;
        double r12674 = r12672 / r12673;
        double r12675 = r12668 * r12674;
        double r12676 = r12669 * r12666;
        double r12677 = -r12676;
        double r12678 = fma(r12666, r12675, r12677);
        return r12678;
}

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.2

    \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
  2. Using strategy rm

  3. Applied flip--0.2

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

  4. Applied associate-*r/0.2

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

  5. Applied associate-/l/0.2

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied associate-*l*0.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied associate-*l*0.4

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))