Average Error: 0.1 → 0.6
Time: 52.2s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{\left(1 - {m}^{3}\right) \cdot m}{\mathsf{fma}\left(\mathsf{fma}\left(m, m, m\right), v, v\right)} - 1\right) \cdot \left(1 - m\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\frac{\left(1 - {m}^{3}\right) \cdot m}{\mathsf{fma}\left(\mathsf{fma}\left(m, m, m\right), v, v\right)} - 1\right) \cdot \left(1 - m\right)
double f(double m, double v) {
        double r999306 = m;
        double r999307 = 1.0;
        double r999308 = r999307 - r999306;
        double r999309 = r999306 * r999308;
        double r999310 = v;
        double r999311 = r999309 / r999310;
        double r999312 = r999311 - r999307;
        double r999313 = r999312 * r999308;
        return r999313;
}
↓
double f(double m, double v) {
        double r999314 = 1.0;
        double r999315 = m;
        double r999316 = 3.0;
        double r999317 = pow(r999315, r999316);
        double r999318 = r999314 - r999317;
        double r999319 = r999318 * r999315;
        double r999320 = fma(r999315, r999315, r999315);
        double r999321 = v;
        double r999322 = fma(r999320, r999321, r999321);
        double r999323 = r999319 / r999322;
        double r999324 = r999323 - r999314;
        double r999325 = r999314 - r999315;
        double r999326 = r999324 * r999325;
        return r999326;
}

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 flip3--0.1

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

  4. Applied associate-*r/0.6

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

  5. Applied associate-/l/0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019151 +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)))