Average Error: 2.4 → 0.4
Time: 50.5s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{i}{i \cdot 2 + 1.0} \cdot \frac{\frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{2}\]
\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}{i \cdot 2 + 1.0} \cdot \frac{\frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{2}
double f(double i) {
        double r2617988 = i;
        double r2617989 = r2617988 * r2617988;
        double r2617990 = r2617989 * r2617989;
        double r2617991 = 2.0;
        double r2617992 = /* ERROR: no posit support in C */;
        double r2617993 = r2617992 * r2617988;
        double r2617994 = r2617993 * r2617993;
        double r2617995 = r2617990 / r2617994;
        double r2617996 = 1.0;
        double r2617997 = /* ERROR: no posit support in C */;
        double r2617998 = r2617994 - r2617997;
        double r2617999 = r2617995 / r2617998;
        return r2617999;
}
↓
double f(double i) {
        double r2618000 = i;
        double r2618001 = 2.0;
        double r2618002 = r2618000 * r2618001;
        double r2618003 = 1.0;
        double r2618004 = r2618002 + r2618003;
        double r2618005 = r2618000 / r2618004;
        double r2618006 = r2618000 / r2618001;
        double r2618007 = r2618002 - r2618003;
        double r2618008 = r2618006 / r2618007;
        double r2618009 = r2618008 / r2618001;
        double r2618010 = r2618005 * r2618009;
        return r2618010;
}

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

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

  4. Applied difference-of-squares2.3

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

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

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

  6. Applied p16-times-frac2.3

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

  7. Simplified0.7

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

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

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

  10. Applied p16-times-frac0.5

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

  11. Applied associate-*r*0.5

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

  12. Simplified0.4

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

  14. Applied associate-*l/0.4

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

  16. Applied p16-times-frac0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(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))))