Average Error: 2.3 → 0.5
Time: 53.2s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i \cdot \frac{\frac{i}{2}}{i \cdot 2 + 1.0}}{2}}{i \cdot 2 - 1.0}\]
\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{i \cdot \frac{\frac{i}{2}}{i \cdot 2 + 1.0}}{2}}{i \cdot 2 - 1.0}
double f(double i) {
        double r2861916 = i;
        double r2861917 = r2861916 * r2861916;
        double r2861918 = r2861917 * r2861917;
        double r2861919 = 2.0;
        double r2861920 = /* ERROR: no posit support in C */;
        double r2861921 = r2861920 * r2861916;
        double r2861922 = r2861921 * r2861921;
        double r2861923 = r2861918 / r2861922;
        double r2861924 = 1.0;
        double r2861925 = /* ERROR: no posit support in C */;
        double r2861926 = r2861922 - r2861925;
        double r2861927 = r2861923 / r2861926;
        return r2861927;
}
↓
double f(double i) {
        double r2861928 = i;
        double r2861929 = 2.0;
        double r2861930 = r2861928 / r2861929;
        double r2861931 = r2861928 * r2861929;
        double r2861932 = 1.0;
        double r2861933 = r2861931 + r2861932;
        double r2861934 = r2861930 / r2861933;
        double r2861935 = r2861928 * r2861934;
        double r2861936 = r2861935 / r2861929;
        double r2861937 = r2861931 - r2861932;
        double r2861938 = r2861936 / r2861937;
        return r2861938;
}

Error

Bits error versus i

Derivation

  1. Initial program 2.3

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

  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-10.8

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

  5. Applied associate-/r*0.5

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

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 +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))))