Average Error: 2.4 → 0.4
Time: 1.4m
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{2}}{i \cdot 2 + 1.0} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}\]
\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{i}{2}}{i \cdot 2 + 1.0} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}
double f(double i) {
        double r2726455 = i;
        double r2726456 = r2726455 * r2726455;
        double r2726457 = r2726456 * r2726456;
        double r2726458 = 2.0;
        double r2726459 = /* ERROR: no posit support in C */;
        double r2726460 = r2726459 * r2726455;
        double r2726461 = r2726460 * r2726460;
        double r2726462 = r2726457 / r2726461;
        double r2726463 = 1.0;
        double r2726464 = /* ERROR: no posit support in C */;
        double r2726465 = r2726461 - r2726464;
        double r2726466 = r2726462 / r2726465;
        return r2726466;
}
↓
double f(double i) {
        double r2726467 = i;
        double r2726468 = 2.0;
        double r2726469 = r2726467 / r2726468;
        double r2726470 = r2726467 * r2726468;
        double r2726471 = 1.0;
        double r2726472 = r2726470 + r2726471;
        double r2726473 = r2726469 / r2726472;
        double r2726474 = r2726470 - r2726471;
        double r2726475 = r2726469 / r2726474;
        double r2726476 = r2726473 * r2726475;
        return r2726476;
}

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. Simplified1.2

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

  4. Applied associate-*r/1.1

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

  6. Applied /p16-rgt-identity-expand1.1

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

  7. Applied associate-/l/1.1

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

  8. Simplified1.1

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

  10. Applied associate-/r*0.8

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

  12. Applied p16-*-un-lft-identity0.8

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

  13. Applied difference-of-squares0.8

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

  14. Applied p16-times-frac0.8

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

  15. Applied p16-times-frac0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 
(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))))