Octave 3.8, jcobi/4, as called

Average Error: 2.4 → 0.4

Time: 28.1s

Precision: 64

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

Math

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

↓

\[\frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]

C

double f(double i) {
        double r1117420 = i;
        double r1117421 = r1117420 * r1117420;
        double r1117422 = r1117421 * r1117421;
        double r1117423 = 2.0;
        double r1117424 = /* ERROR: no posit support in C */;
        double r1117425 = r1117424 * r1117420;
        double r1117426 = r1117425 * r1117425;
        double r1117427 = r1117422 / r1117426;
        double r1117428 = 1.0;
        double r1117429 = /* ERROR: no posit support in C */;
        double r1117430 = r1117426 - r1117429;
        double r1117431 = r1117427 / r1117430;
        return r1117431;
}

↓

double f(double i) {
        double r1117432 = i;
        double r1117433 = 2.0;
        double r1117434 = 1.0;
        double r1117435 = r1117433 / r1117434;
        double r1117436 = r1117432 / r1117435;
        double r1117437 = r1117433 * r1117432;
        double r1117438 = r1117437 + r1117434;
        double r1117439 = r1117436 / r1117438;
        double r1117440 = /*Error: no posit support in C */;
        double r1117441 = /*Error: no posit support in C */;
        double r1117442 = /*Error: no posit support in C */;
        double r1117443 = r1117436 / r1117442;
        double r1117444 = r1117439 * r1117443;
        return r1117444;
}

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. Using strategy rm

3. Applied p16-*-un-lft-identity 2.4

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

4. Applied associate-/r* 2.4

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

5. Simplified 0.9

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

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

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

8. Applied difference-of-squares 0.8

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

9. Applied p16-times-frac 0.4

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

11. Applied introduce-quire 0.4

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

12. Applied insert-quire-sub 0.4

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

13. Final simplification 0.4

    \[\leadsto \frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]

Reproduce

herbie shell --seed 2019156 
(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))))