Average Error: 0.7 → 0.7
Time: 1.0m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]
\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{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r5310387 = beta;
        double r5310388 = alpha;
        double r5310389 = r5310387 - r5310388;
        double r5310390 = r5310388 + r5310387;
        double r5310391 = 2.0;
        double r5310392 = /* ERROR: no posit support in C */;
        double r5310393 = r5310390 + r5310392;
        double r5310394 = r5310389 / r5310393;
        double r5310395 = 1.0;
        double r5310396 = /* ERROR: no posit support in C */;
        double r5310397 = r5310394 + r5310396;
        double r5310398 = r5310397 / r5310392;
        return r5310398;
}
↓
double f(double alpha, double beta) {
        double r5310399 = beta;
        double r5310400 = alpha;
        double r5310401 = r5310399 - r5310400;
        double r5310402 = 2.0;
        double r5310403 = r5310399 + r5310402;
        double r5310404 = r5310400 + r5310403;
        double r5310405 = r5310401 / r5310404;
        double r5310406 = 1.0;
        double r5310407 = r5310405 + r5310406;
        double r5310408 = r5310407 / r5310402;
        return r5310408;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 
(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)))