Average Error: 0.9 → 0.6
Time: 42.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\alpha + \beta}{\beta + \left(\left(i \cdot 2 + \alpha\right) + 2.0\right)} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + i \cdot 2} + 1.0}{2.0}\]
\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}{\beta + \left(\left(i \cdot 2 + \alpha\right) + 2.0\right)} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + i \cdot 2} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r523845 = alpha;
        double r523846 = beta;
        double r523847 = r523845 + r523846;
        double r523848 = r523846 - r523845;
        double r523849 = r523847 * r523848;
        double r523850 = 2.0;
        double r523851 = /* ERROR: no posit support in C */;
        double r523852 = i;
        double r523853 = r523851 * r523852;
        double r523854 = r523847 + r523853;
        double r523855 = r523849 / r523854;
        double r523856 = 2.0;
        double r523857 = /* ERROR: no posit support in C */;
        double r523858 = r523854 + r523857;
        double r523859 = r523855 / r523858;
        double r523860 = 1.0;
        double r523861 = /* ERROR: no posit support in C */;
        double r523862 = r523859 + r523861;
        double r523863 = r523862 / r523857;
        return r523863;
}
↓
double f(double alpha, double beta, double i) {
        double r523864 = alpha;
        double r523865 = beta;
        double r523866 = r523864 + r523865;
        double r523867 = i;
        double r523868 = 2.0;
        double r523869 = r523867 * r523868;
        double r523870 = r523869 + r523864;
        double r523871 = 2.0;
        double r523872 = r523870 + r523871;
        double r523873 = r523865 + r523872;
        double r523874 = r523866 / r523873;
        double r523875 = r523865 - r523864;
        double r523876 = r523865 + r523864;
        double r523877 = r523876 + r523869;
        double r523878 = r523875 / r523877;
        double r523879 = r523874 * r523878;
        double r523880 = 1.0;
        double r523881 = r523879 + r523880;
        double r523882 = r523881 / r523871;
        return r523882;
}

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 *p16-rgt-identity-expand0.9

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

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

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

  5. Applied p16-times-frac0.6

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

  6. Applied p16-times-frac0.6

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

  7. Simplified0.6

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

  8. Simplified0.6

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

  10. Applied *p16-rgt-identity-expand0.6

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

  11. Applied associate-/l*0.6

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

  12. Simplified0.6

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

  14. Applied associate-+r+0.6

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

  16. Applied p16-*-un-lft-identity0.6

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

  17. Applied associate-*r*0.6

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

  18. Simplified0.6

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

  19. Final simplification0.6

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

Reproduce

herbie shell --seed 2019153 
(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)))