Average Error: 2.4 → 0.4
Time: 18.7s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{\frac{2 \cdot i - 1.0}{\frac{i}{2}} \cdot 2}}{2 \cdot i + 1.0}\]
\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}{\frac{2 \cdot i - 1.0}{\frac{i}{2}} \cdot 2}}{2 \cdot i + 1.0}
double f(double i) {
        double r2104972 = i;
        double r2104973 = r2104972 * r2104972;
        double r2104974 = r2104973 * r2104973;
        double r2104975 = 2.0;
        double r2104976 = /* ERROR: no posit support in C */;
        double r2104977 = r2104976 * r2104972;
        double r2104978 = r2104977 * r2104977;
        double r2104979 = r2104974 / r2104978;
        double r2104980 = 1.0;
        double r2104981 = /* ERROR: no posit support in C */;
        double r2104982 = r2104978 - r2104981;
        double r2104983 = r2104979 / r2104982;
        return r2104983;
}
↓
double f(double i) {
        double r2104984 = i;
        double r2104985 = 2.0;
        double r2104986 = r2104985 * r2104984;
        double r2104987 = 1.0;
        double r2104988 = r2104986 - r2104987;
        double r2104989 = r2104984 / r2104985;
        double r2104990 = r2104988 / r2104989;
        double r2104991 = r2104990 * r2104985;
        double r2104992 = r2104984 / r2104991;
        double r2104993 = r2104986 + r2104987;
        double r2104994 = r2104992 / r2104993;
        return r2104994;
}

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 difference-of-sqr-12.4

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

  4. Applied p16-times-frac1.0

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

  5. Applied p16-times-frac1.0

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

  7. Applied associate-*l/1.0

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

  8. Simplified0.6

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

  10. Applied associate-/l*0.4

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

  12. Applied associate-/l/0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 +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))))