Average Error: 2.3 → 0.4
Time: 58.7s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}\]
\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 r3676356 = i;
        double r3676357 = r3676356 * r3676356;
        double r3676358 = r3676357 * r3676357;
        double r3676359 = 2.0;
        double r3676360 = /* ERROR: no posit support in C */;
        double r3676361 = r3676360 * r3676356;
        double r3676362 = r3676361 * r3676361;
        double r3676363 = r3676358 / r3676362;
        double r3676364 = 1.0;
        double r3676365 = /* ERROR: no posit support in C */;
        double r3676366 = r3676362 - r3676365;
        double r3676367 = r3676363 / r3676366;
        return r3676367;
}
↓
double f(double i) {
        double r3676368 = i;
        double r3676369 = 2.0;
        double r3676370 = r3676368 / r3676369;
        double r3676371 = r3676368 * r3676369;
        double r3676372 = 1.0;
        double r3676373 = r3676371 - r3676372;
        double r3676374 = r3676373 / r3676370;
        double r3676375 = r3676370 / r3676374;
        double r3676376 = r3676371 + r3676372;
        double r3676377 = r3676375 / r3676376;
        return r3676377;
}

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

    \[\leadsto \color{blue}{\frac{\left(\frac{i}{\left(2\right)}\right)}{\left(\frac{\left(\left(i \cdot \left(2\right)\right) - \left(1.0\right)\right)}{\left(\frac{\left(\frac{i}{\left(2\right)}\right)}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(1.0\right)}\right)}\right)}\right)}}\]
  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 2019135 +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))))