Average Error: 0.1 → 0.1
Time: 22.1s
Precision: 64
\[0 \lt m \land 0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(1 - \sqrt{m}\right) \cdot \mathsf{fma}\left(\sqrt{m}, \frac{\left(1 - m\right) \cdot m}{v} - 1, \frac{\left(1 - m\right) \cdot m}{v} - 1\right)\]
\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}, \frac{\left(1 - m\right) \cdot m}{v} - 1, \frac{\left(1 - m\right) \cdot m}{v} - 1\right)
double f(double m, double v) {
        double r1128415 = m;
        double r1128416 = 1.0;
        double r1128417 = r1128416 - r1128415;
        double r1128418 = r1128415 * r1128417;
        double r1128419 = v;
        double r1128420 = r1128418 / r1128419;
        double r1128421 = r1128420 - r1128416;
        double r1128422 = r1128421 * r1128417;
        return r1128422;
}
↓
double f(double m, double v) {
        double r1128423 = 1.0;
        double r1128424 = m;
        double r1128425 = sqrt(r1128424);
        double r1128426 = r1128423 - r1128425;
        double r1128427 = r1128423 - r1128424;
        double r1128428 = r1128427 * r1128424;
        double r1128429 = v;
        double r1128430 = r1128428 / r1128429;
        double r1128431 = r1128430 - r1128423;
        double r1128432 = fma(r1128425, r1128431, r1128431);
        double r1128433 = r1128426 * r1128432;
        return r1128433;
}

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 add-sqr-sqrt0.1

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

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

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

  5. Applied difference-of-squares0.1

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

  6. Applied associate-*r*0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +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)))