Average Error: 0.7 → 0.8
Time: 34.1s
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 r871766 = beta;
        double r871767 = alpha;
        double r871768 = r871766 - r871767;
        double r871769 = r871767 + r871766;
        double r871770 = 2.0;
        double r871771 = /* ERROR: no posit support in C */;
        double r871772 = r871769 + r871771;
        double r871773 = r871768 / r871772;
        double r871774 = 1.0;
        double r871775 = /* ERROR: no posit support in C */;
        double r871776 = r871773 + r871775;
        double r871777 = r871776 / r871771;
        return r871777;
}
↓
double f(double alpha, double beta) {
        double r871778 = 1.0;
        double r871779 = alpha;
        double r871780 = beta;
        double r871781 = r871779 + r871780;
        double r871782 = 2.0;
        double r871783 = r871781 + r871782;
        double r871784 = r871780 - r871779;
        double r871785 = r871783 / r871784;
        double r871786 = r871778 / r871785;
        double r871787 = r871786 + r871778;
        double r871788 = r871787 / r871782;
        return r871788;
}

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*0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019155 
(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)))