Average Error: 0.7 → 0.8
Time: 24.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{1.0}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta - \alpha}} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{1.0}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r2534342 = beta;
        double r2534343 = alpha;
        double r2534344 = r2534342 - r2534343;
        double r2534345 = r2534343 + r2534342;
        double r2534346 = 2.0;
        double r2534347 = /* ERROR: no posit support in C */;
        double r2534348 = r2534345 + r2534347;
        double r2534349 = r2534344 / r2534348;
        double r2534350 = 1.0;
        double r2534351 = /* ERROR: no posit support in C */;
        double r2534352 = r2534349 + r2534351;
        double r2534353 = r2534352 / r2534347;
        return r2534353;
}
↓
double f(double alpha, double beta) {
        double r2534354 = 1.0;
        double r2534355 = beta;
        double r2534356 = 2.0;
        double r2534357 = r2534355 + r2534356;
        double r2534358 = alpha;
        double r2534359 = r2534357 + r2534358;
        double r2534360 = r2534355 - r2534358;
        double r2534361 = r2534359 / r2534360;
        double r2534362 = r2534354 / r2534361;
        double r2534363 = r2534362 + r2534354;
        double r2534364 = r2534363 / r2534356;
        return r2534364;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.7

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  4. Using strategy rm

  5. Applied p16-*-un-lft-identity0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  6. Applied associate-/l*0.8

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  7. Using strategy rm

  8. Applied +-commutative0.8

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\beta}{\left(2.0\right)}\right)}{\alpha}\right)}}{\left(\beta - \alpha\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  9. Final simplification0.8

    \[\leadsto \frac{\frac{1.0}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta - \alpha}} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))