Average Error: 1.0 → 0.7
Time: 32.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r928798 = alpha;
        double r928799 = beta;
        double r928800 = r928798 + r928799;
        double r928801 = r928799 - r928798;
        double r928802 = r928800 * r928801;
        double r928803 = 2.0;
        double r928804 = /* ERROR: no posit support in C */;
        double r928805 = i;
        double r928806 = r928804 * r928805;
        double r928807 = r928800 + r928806;
        double r928808 = r928802 / r928807;
        double r928809 = 2.0;
        double r928810 = /* ERROR: no posit support in C */;
        double r928811 = r928807 + r928810;
        double r928812 = r928808 / r928811;
        double r928813 = 1.0;
        double r928814 = /* ERROR: no posit support in C */;
        double r928815 = r928812 + r928814;
        double r928816 = r928815 / r928810;
        return r928816;
}
↓
double f(double alpha, double beta, double i) {
        double r928817 = alpha;
        double r928818 = beta;
        double r928819 = r928817 + r928818;
        double r928820 = 2.0;
        double r928821 = i;
        double r928822 = r928820 * r928821;
        double r928823 = r928819 + r928822;
        double r928824 = r928818 - r928817;
        double r928825 = r928823 / r928824;
        double r928826 = r928819 / r928825;
        double r928827 = 2.0;
        double r928828 = r928823 + r928827;
        double r928829 = r928826 / r928828;
        double r928830 = 1.0;
        double r928831 = r928829 + r928830;
        double r928832 = r928831 / r928827;
        return r928832;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 1.0

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied associate-/l*0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  4. Final simplification0.7

    \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))