Average Error: 0.7 → 0.7
Time: 19.5s
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 r3124652 = beta;
        double r3124653 = alpha;
        double r3124654 = r3124652 - r3124653;
        double r3124655 = r3124653 + r3124652;
        double r3124656 = 2.0;
        double r3124657 = /* ERROR: no posit support in C */;
        double r3124658 = r3124655 + r3124657;
        double r3124659 = r3124654 / r3124658;
        double r3124660 = 1.0;
        double r3124661 = /* ERROR: no posit support in C */;
        double r3124662 = r3124659 + r3124661;
        double r3124663 = r3124662 / r3124657;
        return r3124663;
}
↓
double f(double alpha, double beta) {
        double r3124664 = beta;
        double r3124665 = alpha;
        double r3124666 = r3124664 - r3124665;
        double r3124667 = 2.0;
        double r3124668 = r3124664 + r3124667;
        double r3124669 = r3124665 + r3124668;
        double r3124670 = r3124666 / r3124669;
        double r3124671 = 1.0;
        double r3124672 = r3124670 + r3124671;
        double r3124673 = r3124672 / r3124667;
        return r3124673;
}

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 2019128 
(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)))