Average Error: 2.4 → 0.4
Time: 49.3s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{i}{\left(2 \cdot i + 1.0\right) \cdot \frac{2}{1.0}} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{\left(\mathsf{qms}\left(\left(\left(2 \cdot i\right)\right), 1.0, 1.0\right)\right)}\]
\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}{\left(2 \cdot i + 1.0\right) \cdot \frac{2}{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)}
double f(double i) {
        double r1182114 = i;
        double r1182115 = r1182114 * r1182114;
        double r1182116 = r1182115 * r1182115;
        double r1182117 = 2.0;
        double r1182118 = /* ERROR: no posit support in C */;
        double r1182119 = r1182118 * r1182114;
        double r1182120 = r1182119 * r1182119;
        double r1182121 = r1182116 / r1182120;
        double r1182122 = 1.0;
        double r1182123 = /* ERROR: no posit support in C */;
        double r1182124 = r1182120 - r1182123;
        double r1182125 = r1182121 / r1182124;
        return r1182125;
}
↓
double f(double i) {
        double r1182126 = i;
        double r1182127 = 2.0;
        double r1182128 = r1182127 * r1182126;
        double r1182129 = 1.0;
        double r1182130 = r1182128 + r1182129;
        double r1182131 = r1182127 / r1182129;
        double r1182132 = r1182130 * r1182131;
        double r1182133 = r1182126 / r1182132;
        double r1182134 = r1182126 / r1182131;
        double r1182135 = /*Error: no posit support in C */;
        double r1182136 = /*Error: no posit support in C */;
        double r1182137 = /*Error: no posit support in C */;
        double r1182138 = r1182134 / r1182137;
        double r1182139 = r1182133 * r1182138;
        return r1182139;
}

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-identity2.4

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

  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. Simplified0.9

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

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

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

  8. Applied difference-of-squares0.8

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

  9. Applied p16-times-frac0.4

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

  11. Applied *p16-rgt-identity-expand0.4

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

  12. Applied introduce-quire0.4

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

  13. Applied insert-quire-fdp-sub0.4

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

  15. Applied associate-/l/0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 +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))))