Average Error: 0.9 → 0.6
Time: 38.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) + 2.0}{\beta - \alpha} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1.0}{2.0}\]
\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{\alpha + \beta}{\frac{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) + 2.0}{\beta - \alpha} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r3205857 = alpha;
        double r3205858 = beta;
        double r3205859 = r3205857 + r3205858;
        double r3205860 = r3205858 - r3205857;
        double r3205861 = r3205859 * r3205860;
        double r3205862 = 2.0;
        double r3205863 = /* ERROR: no posit support in C */;
        double r3205864 = i;
        double r3205865 = r3205863 * r3205864;
        double r3205866 = r3205859 + r3205865;
        double r3205867 = r3205861 / r3205866;
        double r3205868 = 2.0;
        double r3205869 = /* ERROR: no posit support in C */;
        double r3205870 = r3205866 + r3205869;
        double r3205871 = r3205867 / r3205870;
        double r3205872 = 1.0;
        double r3205873 = /* ERROR: no posit support in C */;
        double r3205874 = r3205871 + r3205873;
        double r3205875 = r3205874 / r3205869;
        return r3205875;
}
↓
double f(double alpha, double beta, double i) {
        double r3205876 = alpha;
        double r3205877 = beta;
        double r3205878 = r3205876 + r3205877;
        double r3205879 = 2.0;
        double r3205880 = i;
        double r3205881 = r3205879 * r3205880;
        double r3205882 = r3205877 + r3205881;
        double r3205883 = r3205876 + r3205882;
        double r3205884 = 2.0;
        double r3205885 = r3205883 + r3205884;
        double r3205886 = r3205877 - r3205876;
        double r3205887 = r3205885 / r3205886;
        double r3205888 = r3205878 + r3205881;
        double r3205889 = r3205887 * r3205888;
        double r3205890 = r3205878 / r3205889;
        double r3205891 = 1.0;
        double r3205892 = r3205890 + r3205891;
        double r3205893 = r3205892 / r3205884;
        return r3205893;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

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

  3. Applied associate-/l*0.6

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

  5. Applied associate-/r/0.6

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

  6. Applied associate-/l*0.6

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

  8. Applied associate-+l+0.5

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

  10. Applied associate-/l/0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(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)))