Average Error: 0.7 → 0.7
Time: 31.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{1.0}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]
\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(\alpha + \beta\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r2860763 = beta;
        double r2860764 = alpha;
        double r2860765 = r2860763 - r2860764;
        double r2860766 = r2860764 + r2860763;
        double r2860767 = 2.0;
        double r2860768 = /* ERROR: no posit support in C */;
        double r2860769 = r2860766 + r2860768;
        double r2860770 = r2860765 / r2860769;
        double r2860771 = 1.0;
        double r2860772 = /* ERROR: no posit support in C */;
        double r2860773 = r2860770 + r2860772;
        double r2860774 = r2860773 / r2860768;
        return r2860774;
}
↓
double f(double alpha, double beta) {
        double r2860775 = 1.0;
        double r2860776 = alpha;
        double r2860777 = beta;
        double r2860778 = r2860776 + r2860777;
        double r2860779 = 2.0;
        double r2860780 = r2860778 + r2860779;
        double r2860781 = r2860777 - r2860776;
        double r2860782 = r2860780 / r2860781;
        double r2860783 = r2860775 / r2860782;
        double r2860784 = r2860783 + r2860775;
        double r2860785 = r2860784 / r2860779;
        return r2860785;
}

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 p16-*-un-lft-identity0.7

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

  4. Applied associate-/l*0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 
(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)))