Octave 3.8, jcobi/4, as called

Average Error: 2.4 → 0.4

Time: 1.0m

Precision: 64

\[i \gt \left(0\right)\]

Math

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

↓

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

C

double f(double i) {
        double r2676461 = i;
        double r2676462 = r2676461 * r2676461;
        double r2676463 = r2676462 * r2676462;
        double r2676464 = 2.0;
        double r2676465 = /* ERROR: no posit support in C */;
        double r2676466 = r2676465 * r2676461;
        double r2676467 = r2676466 * r2676466;
        double r2676468 = r2676463 / r2676467;
        double r2676469 = 1.0;
        double r2676470 = /* ERROR: no posit support in C */;
        double r2676471 = r2676467 - r2676470;
        double r2676472 = r2676468 / r2676471;
        return r2676472;
}

↓

double f(double i) {
        double r2676473 = i;
        double r2676474 = 2.0;
        double r2676475 = r2676474 * r2676474;
        double r2676476 = r2676473 / r2676475;
        double r2676477 = r2676473 * r2676474;
        double r2676478 = 1.0;
        double r2676479 = r2676477 + r2676478;
        double r2676480 = r2676473 / r2676479;
        double r2676481 = r2676477 - r2676478;
        double r2676482 = r2676480 / r2676481;
        double r2676483 = r2676476 * r2676482;
        return r2676483;
}

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. Simplified 1.2

   \[\leadsto \color{blue}{i \cdot \left(\frac{i}{\left(\left(2\right) \cdot \left(\left(2\right) \cdot \left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) - \left(1.0\right)\right)\right)\right)}\right)}\]
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-expand 0.9

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

9. Applied p16-times-frac 1.0

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

10. Applied p16-times-frac 0.9

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

11. Simplified 0.9

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

12. Simplified 0.9

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

14. Applied p16-*-un-lft-identity 0.9

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

15. Applied difference-of-squares 0.8

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

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 simplification 0.4

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

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))))