Average Error: 0.8 → 0.8
Time: 31.8s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\mathsf{qma}\left(\left(1.0\right), \left(\beta - \alpha\right), \left(\frac{1.0}{2.0 + \left(\alpha + \beta\right)}\right)\right)\right)}{2.0}\]
\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{\left(\mathsf{qma}\left(\left(1.0\right), \left(\beta - \alpha\right), \left(\frac{1.0}{2.0 + \left(\alpha + \beta\right)}\right)\right)\right)}{2.0}
double f(double alpha, double beta) {
        double r3636869 = beta;
        double r3636870 = alpha;
        double r3636871 = r3636869 - r3636870;
        double r3636872 = r3636870 + r3636869;
        double r3636873 = 2.0;
        double r3636874 = /* ERROR: no posit support in C */;
        double r3636875 = r3636872 + r3636874;
        double r3636876 = r3636871 / r3636875;
        double r3636877 = 1.0;
        double r3636878 = /* ERROR: no posit support in C */;
        double r3636879 = r3636876 + r3636878;
        double r3636880 = r3636879 / r3636874;
        return r3636880;
}
↓
double f(double alpha, double beta) {
        double r3636881 = 1.0;
        double r3636882 = /*Error: no posit support in C */;
        double r3636883 = beta;
        double r3636884 = alpha;
        double r3636885 = r3636883 - r3636884;
        double r3636886 = 2.0;
        double r3636887 = r3636884 + r3636883;
        double r3636888 = r3636886 + r3636887;
        double r3636889 = r3636881 / r3636888;
        double r3636890 = /*Error: no posit support in C */;
        double r3636891 = /*Error: no posit support in C */;
        double r3636892 = r3636891 / r3636886;
        return r3636892;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.8

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

  3. Applied associate-+l+0.8

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

  5. Applied p16-*-un-lft-identity0.8

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

  6. Applied associate-/r*0.8

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

  7. Simplified0.8

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

  9. Applied p16-*-un-lft-identity0.8

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

  10. Applied *p16-rgt-identity-expand0.8

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

  11. Applied p16-times-frac0.9

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

  12. Applied introduce-quire0.9

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

  13. Applied insert-quire-fdp-add0.8

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

  14. Simplified0.8

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

  15. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))