Average Error: 0.2 → 0.2
Time: 3.5s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\mathsf{fma}\left(1, \frac{{m}^{1}}{\frac{v}{m}}, -\mathsf{fma}\left(1, m, \frac{{m}^{3}}{v}\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(1, \frac{{m}^{1}}{\frac{v}{m}}, -\mathsf{fma}\left(1, m, \frac{{m}^{3}}{v}\right)\right)
double f(double m, double v) {
        double r9102 = m;
        double r9103 = 1.0;
        double r9104 = r9103 - r9102;
        double r9105 = r9102 * r9104;
        double r9106 = v;
        double r9107 = r9105 / r9106;
        double r9108 = r9107 - r9103;
        double r9109 = r9108 * r9102;
        return r9109;
}
↓
double f(double m, double v) {
        double r9110 = 1.0;
        double r9111 = m;
        double r9112 = 1.0;
        double r9113 = pow(r9111, r9112);
        double r9114 = v;
        double r9115 = r9114 / r9111;
        double r9116 = r9113 / r9115;
        double r9117 = 3.0;
        double r9118 = pow(r9111, r9117);
        double r9119 = r9118 / r9114;
        double r9120 = fma(r9110, r9111, r9119);
        double r9121 = -r9120;
        double r9122 = fma(r9110, r9116, r9121);
        return r9122;
}

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. Taylor expanded around 0 6.3

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

  3. Simplified6.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{{m}^{2}}{v}, -\mathsf{fma}\left(1, m, \frac{{m}^{3}}{v}\right)\right)}\]
  4. Using strategy rm

  5. Applied sqr-pow6.3

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

  6. Applied associate-/l*0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +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))