Average Error: 0.1 → 0.1
Time: 22.2s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\left(\frac{\left(m \cdot m\right) \cdot m}{v} - \frac{m \cdot m}{v}\right) + m\right) + \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\left(\frac{\left(m \cdot m\right) \cdot m}{v} - \frac{m \cdot m}{v}\right) + m\right) + \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)
double f(double m, double v) {
        double r655207 = m;
        double r655208 = 1.0;
        double r655209 = r655208 - r655207;
        double r655210 = r655207 * r655209;
        double r655211 = v;
        double r655212 = r655210 / r655211;
        double r655213 = r655212 - r655208;
        double r655214 = r655213 * r655209;
        return r655214;
}
↓
double f(double m, double v) {
        double r655215 = m;
        double r655216 = r655215 * r655215;
        double r655217 = r655216 * r655215;
        double r655218 = v;
        double r655219 = r655217 / r655218;
        double r655220 = r655216 / r655218;
        double r655221 = r655219 - r655220;
        double r655222 = r655221 + r655215;
        double r655223 = 1.0;
        double r655224 = r655223 - r655215;
        double r655225 = r655215 / r655218;
        double r655226 = -1.0;
        double r655227 = fma(r655224, r655225, r655226);
        double r655228 = r655222 + r655227;
        return r655228;
}

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. Simplified0.1

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

  4. Applied sub-neg0.1

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

  5. Applied distribute-rgt-in0.1

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

  6. Taylor expanded around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))