Average Error: 2.4 → 0.4
Time: 22.6s
Precision: 64
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}\]
double f(double i) {
        double r2088867 = i;
        double r2088868 = r2088867 * r2088867;
        double r2088869 = r2088868 * r2088868;
        double r2088870 = 2.0;
        double r2088871 = r2088870 * r2088867;
        double r2088872 = r2088871 * r2088871;
        double r2088873 = r2088869 / r2088872;
        double r2088874 = 1.0;
        double r2088875 = r2088872 - r2088874;
        double r2088876 = r2088873 / r2088875;
        return r2088876;
}
↓
double f(double i) {
        double r2088877 = i;
        double r2088878 = 2.0;
        double r2088879 = r2088877 / r2088878;
        double r2088880 = r2088877 * r2088878;
        double r2088881 = 1.0;
        double r2088882 = r2088880 - r2088881;
        double r2088883 = r2088879 / r2088882;
        double r2088884 = r2088879 * r2088883;
        double r2088885 = r2088880 + r2088881;
        double r2088886 = r2088884 / r2088885;
        return r2088886;
}


\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}
↓
\frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}

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. Simplified2.4

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

  4. Applied p16-times-frac1.2

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

  6. Applied associate-*r/1.2

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

  7. Applied associate-*l/1.1

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

  8. Simplified0.9

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

  10. Applied difference-of-sqr-10.8

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

  11. Applied p16-times-frac0.4

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

  13. Applied associate-*l/0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +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))))