Average Error: 2.4 → 0.4
Time: 48.1s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{i}{\left(2 \cdot i + 1.0\right) \cdot \frac{2}{1.0}} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]
\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{i}{\left(2 \cdot i + 1.0\right) \cdot \frac{2}{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 r958679 = i;
        double r958680 = r958679 * r958679;
        double r958681 = r958680 * r958680;
        double r958682 = 2.0;
        double r958683 = /* ERROR: no posit support in C */;
        double r958684 = r958683 * r958679;
        double r958685 = r958684 * r958684;
        double r958686 = r958681 / r958685;
        double r958687 = 1.0;
        double r958688 = /* ERROR: no posit support in C */;
        double r958689 = r958685 - r958688;
        double r958690 = r958686 / r958689;
        return r958690;
}
↓
double f(double i) {
        double r958691 = i;
        double r958692 = 2.0;
        double r958693 = r958692 * r958691;
        double r958694 = 1.0;
        double r958695 = r958693 + r958694;
        double r958696 = r958692 / r958694;
        double r958697 = r958695 * r958696;
        double r958698 = r958691 / r958697;
        double r958699 = r958691 / r958696;
        double r958700 = /*Error: no posit support in C */;
        double r958701 = /*Error: no posit support in C */;
        double r958702 = /*Error: no posit support in C */;
        double r958703 = r958699 / r958702;
        double r958704 = r958698 * r958703;
        return r958704;
}

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-lft-identity-expand2.4

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

  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-lft-identity-expand0.9

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

  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 p16-*-un-lft-identity0.4

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

  12. Applied introduce-quire0.4

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

  13. Applied insert-quire-fdp-sub0.4

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

  15. Applied associate-/l/0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(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))))