Average Error: 2.4 → 0.4
Time: 28.8s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]
\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}
↓
\frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}
double f(double i) {
        double r1232674 = i;
        double r1232675 = r1232674 * r1232674;
        double r1232676 = r1232675 * r1232675;
        double r1232677 = 2.0;
        double r1232678 = /* ERROR: no posit support in C */;
        double r1232679 = r1232678 * r1232674;
        double r1232680 = r1232679 * r1232679;
        double r1232681 = r1232676 / r1232680;
        double r1232682 = 1.0;
        double r1232683 = /* ERROR: no posit support in C */;
        double r1232684 = r1232680 - r1232683;
        double r1232685 = r1232681 / r1232684;
        return r1232685;
}
↓
double f(double i) {
        double r1232686 = i;
        double r1232687 = 2.0;
        double r1232688 = 1.0;
        double r1232689 = r1232687 / r1232688;
        double r1232690 = r1232686 / r1232689;
        double r1232691 = r1232687 * r1232686;
        double r1232692 = r1232691 + r1232688;
        double r1232693 = r1232690 / r1232692;
        double r1232694 = /*Error: no posit support in C */;
        double r1232695 = /*Error: no posit support in C */;
        double r1232696 = /*Error: no posit support in C */;
        double r1232697 = r1232690 / r1232696;
        double r1232698 = r1232693 * r1232697;
        return r1232698;
}

Error

Bits error versus i

Derivation

  1. Initial program 2.4

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

  3. Applied p16-*-un-lft-identity2.4

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

  4. Applied associate-/r*2.4

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

  5. Simplified0.9

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

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

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

  8. Applied difference-of-squares0.8

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

  9. Applied p16-times-frac0.4

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

  11. Applied introduce-quire0.4

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

  12. Applied insert-quire-sub0.4

    \[\leadsto \left(\frac{\left(\frac{i}{\left(\frac{\left(2\right)}{\left(1.0\right)}\right)}\right)}{\left(\frac{\left(\left(2\right) \cdot i\right)}{\left(1.0\right)}\right)}\right) \cdot \left(\frac{\left(\frac{i}{\left(\frac{\left(2\right)}{\left(1.0\right)}\right)}\right)}{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(\left(2\right) \cdot i\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}}\right)\]

  13. Final simplification0.4

    \[\leadsto \frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (>.p16 i (real->posit16 0)))
  (/.p16 (/.p16 (*.p16 (*.p16 i i) (*.p16 i i)) (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i))) (-.p16 (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i)) (real->posit16 1.0))))