Average Error: 0.7 → 0.7
Time: 19.6s
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 r2240683 = beta;
        double r2240684 = alpha;
        double r2240685 = r2240683 - r2240684;
        double r2240686 = r2240684 + r2240683;
        double r2240687 = 2.0;
        double r2240688 = /* ERROR: no posit support in C */;
        double r2240689 = r2240686 + r2240688;
        double r2240690 = r2240685 / r2240689;
        double r2240691 = 1.0;
        double r2240692 = /* ERROR: no posit support in C */;
        double r2240693 = r2240690 + r2240692;
        double r2240694 = r2240693 / r2240688;
        return r2240694;
}
↓
double f(double alpha, double beta) {
        double r2240695 = beta;
        double r2240696 = alpha;
        double r2240697 = r2240695 - r2240696;
        double r2240698 = 2.0;
        double r2240699 = r2240695 + r2240698;
        double r2240700 = r2240696 + r2240699;
        double r2240701 = r2240697 / r2240700;
        double r2240702 = 1.0;
        double r2240703 = r2240701 + r2240702;
        double r2240704 = r2240703 / r2240698;
        return r2240704;
}

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)))