Average Error: 0.1 → 0.1
Time: 1.5m
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(1 - \sqrt{m}\right) \cdot \mathsf{fma}\left(\sqrt{m}, \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right), \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(1 - \sqrt{m}\right) \cdot \mathsf{fma}\left(\sqrt{m}, \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right), \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)\right)
double f(double m, double v) {
        double r1649339 = m;
        double r1649340 = 1.0;
        double r1649341 = r1649340 - r1649339;
        double r1649342 = r1649339 * r1649341;
        double r1649343 = v;
        double r1649344 = r1649342 / r1649343;
        double r1649345 = r1649344 - r1649340;
        double r1649346 = r1649345 * r1649341;
        return r1649346;
}
↓
double f(double m, double v) {
        double r1649347 = 1.0;
        double r1649348 = m;
        double r1649349 = sqrt(r1649348);
        double r1649350 = r1649347 - r1649349;
        double r1649351 = r1649347 - r1649348;
        double r1649352 = v;
        double r1649353 = r1649348 / r1649352;
        double r1649354 = -1.0;
        double r1649355 = fma(r1649351, r1649353, r1649354);
        double r1649356 = fma(r1649349, r1649355, r1649355);
        double r1649357 = r1649350 * r1649356;
        return r1649357;
}

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

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

  4. Applied add-sqr-sqrt0.1

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

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

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

  6. Applied difference-of-squares0.1

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

  7. Applied associate-*r*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 +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)))