Average Error: 0.7 → 0.7
Time: 19.9s
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 r1342915 = beta;
        double r1342916 = alpha;
        double r1342917 = r1342915 - r1342916;
        double r1342918 = r1342916 + r1342915;
        double r1342919 = 2.0;
        double r1342920 = /* ERROR: no posit support in C */;
        double r1342921 = r1342918 + r1342920;
        double r1342922 = r1342917 / r1342921;
        double r1342923 = 1.0;
        double r1342924 = /* ERROR: no posit support in C */;
        double r1342925 = r1342922 + r1342924;
        double r1342926 = r1342925 / r1342920;
        return r1342926;
}
↓
double f(double alpha, double beta) {
        double r1342927 = beta;
        double r1342928 = alpha;
        double r1342929 = r1342927 - r1342928;
        double r1342930 = 2.0;
        double r1342931 = r1342927 + r1342930;
        double r1342932 = r1342928 + r1342931;
        double r1342933 = r1342929 / r1342932;
        double r1342934 = 1.0;
        double r1342935 = r1342933 + r1342934;
        double r1342936 = r1342935 / r1342930;
        return r1342936;
}

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 2019124 
(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)))