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 r3423496 = i;
        double r3423497 = r3423496 * r3423496;
        double r3423498 = r3423497 * r3423497;
        double r3423499 = 2.0;
        double r3423500 = /* ERROR: no posit support in C */;
        double r3423501 = r3423500 * r3423496;
        double r3423502 = r3423501 * r3423501;
        double r3423503 = r3423498 / r3423502;
        double r3423504 = 1.0;
        double r3423505 = /* ERROR: no posit support in C */;
        double r3423506 = r3423502 - r3423505;
        double r3423507 = r3423503 / r3423506;
        return r3423507;
}

↓

double f(double i) {
        double r3423508 = i;
        double r3423509 = 2.0;
        double r3423510 = r3423509 * r3423509;
        double r3423511 = r3423508 / r3423510;
        double r3423512 = r3423508 * r3423509;
        double r3423513 = 1.0;
        double r3423514 = r3423512 + r3423513;
        double r3423515 = r3423508 / r3423514;
        double r3423516 = r3423512 - r3423513;
        double r3423517 = r3423515 / r3423516;
        double r3423518 = r3423511 * r3423517;
        return r3423518;
}

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 
(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))))