Average Error: 2.4 → 0.4
Time: 55.6s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}\]
\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{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}
double f(double i) {
        double r6928719 = i;
        double r6928720 = r6928719 * r6928719;
        double r6928721 = r6928720 * r6928720;
        double r6928722 = 2.0;
        double r6928723 = /* ERROR: no posit support in C */;
        double r6928724 = r6928723 * r6928719;
        double r6928725 = r6928724 * r6928724;
        double r6928726 = r6928721 / r6928725;
        double r6928727 = 1.0;
        double r6928728 = /* ERROR: no posit support in C */;
        double r6928729 = r6928725 - r6928728;
        double r6928730 = r6928726 / r6928729;
        return r6928730;
}
↓
double f(double i) {
        double r6928731 = i;
        double r6928732 = 2.0;
        double r6928733 = r6928731 / r6928732;
        double r6928734 = r6928731 * r6928732;
        double r6928735 = 1.0;
        double r6928736 = r6928734 - r6928735;
        double r6928737 = r6928736 / r6928733;
        double r6928738 = r6928733 / r6928737;
        double r6928739 = r6928734 + r6928735;
        double r6928740 = r6928738 / r6928739;
        return r6928740;
}

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. Simplified0.9

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

  4. Applied difference-of-sqr-10.9

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

  5. Applied associate-/r*0.6

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

  7. Applied associate-*r/0.5

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

  9. Applied associate-/l*0.5

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

  11. Applied associate-/r/0.5

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

  12. Applied associate-/r*0.4

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

  13. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}\]

Reproduce

herbie shell --seed 2019130 
(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))))