Average Error: 0.1 → 0.1
Time: 11.9s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(-m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(-m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)\right)
double f(double m, double v) {
        double r16310 = m;
        double r16311 = 1.0;
        double r16312 = r16311 - r16310;
        double r16313 = r16310 * r16312;
        double r16314 = v;
        double r16315 = r16313 / r16314;
        double r16316 = r16315 - r16311;
        double r16317 = r16316 * r16312;
        return r16317;
}
↓
double f(double m, double v) {
        double r16318 = 1.0;
        double r16319 = m;
        double r16320 = r16318 - r16319;
        double r16321 = r16319 * r16320;
        double r16322 = v;
        double r16323 = r16321 / r16322;
        double r16324 = r16323 - r16318;
        double r16325 = r16318 * r16324;
        double r16326 = r16320 / r16322;
        double r16327 = -r16318;
        double r16328 = fma(r16319, r16326, r16327);
        double r16329 = r16319 * r16328;
        double r16330 = -r16329;
        double r16331 = r16325 + r16330;
        return r16331;
}

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 sub-neg0.1

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

  4. Applied distribute-lft-in0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

    \[\leadsto 1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \color{blue}{\left(-m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}\]
  7. Using strategy rm

  8. Applied pow10.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-down0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))