Average Error: 2.4 → 0.4
Time: 58.9s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\left(\frac{i}{\left(\left(2\right) \cdot \left(2\right)\right)}\right) \cdot \left(\frac{\left(\frac{i}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(1.0\right)}\right)}\right)}{\left(\left(i \cdot \left(2\right)\right) - \left(1.0\right)\right)}\right)\]
\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)}
↓
\left(\frac{i}{\left(\left(2\right) \cdot \left(2\right)\right)}\right) \cdot \left(\frac{\left(\frac{i}{\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)
double f(double i) {
        double r2676460 = i;
        double r2676461 = r2676460 * r2676460;
        double r2676462 = r2676461 * r2676461;
        double r2676463 = 2.0;
        double r2676464 = /* ERROR: no posit support in C */;
        double r2676465 = r2676464 * r2676460;
        double r2676466 = r2676465 * r2676465;
        double r2676467 = r2676462 / r2676466;
        double r2676468 = 1.0;
        double r2676469 = /* ERROR: no posit support in C */;
        double r2676470 = r2676466 - r2676469;
        double r2676471 = r2676467 / r2676470;
        return r2676471;
}
↓
double f(double i) {
        double r2676472 = i;
        double r2676473 = 2.0;
        double r2676474 = /* ERROR: no posit support in C */;
        double r2676475 = r2676474 * r2676474;
        double r2676476 = r2676472 / r2676475;
        double r2676477 = r2676472 * r2676474;
        double r2676478 = 1.0;
        double r2676479 = /* ERROR: no posit support in C */;
        double r2676480 = r2676477 + r2676479;
        double r2676481 = r2676472 / r2676480;
        double r2676482 = r2676477 - r2676479;
        double r2676483 = r2676481 / r2676482;
        double r2676484 = r2676476 * r2676483;
        return r2676484;
}

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 associate-/r*0.9

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

  8. Applied *p16-rgt-identity-expand0.9

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

  9. Applied p16-times-frac1.0

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

  10. Applied p16-times-frac0.9

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

  11. Simplified0.9

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

  12. Simplified0.9

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

  14. Applied p16-*-un-lft-identity0.9

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

  15. Applied difference-of-squares0.8

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

  16. Applied associate-/r*0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +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))))