Average Error: 2.4 → 0.4
Time: 31.6s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{1.0}{\frac{2 \cdot i + 1.0}{i}}}{\frac{2}{1.0}} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i - 1.0}\]
\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{1.0}{\frac{2 \cdot i + 1.0}{i}}}{\frac{2}{1.0}} \cdot \frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i - 1.0}
double f(double i) {
        double r1032103 = i;
        double r1032104 = r1032103 * r1032103;
        double r1032105 = r1032104 * r1032104;
        double r1032106 = 2.0;
        double r1032107 = /* ERROR: no posit support in C */;
        double r1032108 = r1032107 * r1032103;
        double r1032109 = r1032108 * r1032108;
        double r1032110 = r1032105 / r1032109;
        double r1032111 = 1.0;
        double r1032112 = /* ERROR: no posit support in C */;
        double r1032113 = r1032109 - r1032112;
        double r1032114 = r1032110 / r1032113;
        return r1032114;
}
↓
double f(double i) {
        double r1032115 = 1.0;
        double r1032116 = 2.0;
        double r1032117 = i;
        double r1032118 = r1032116 * r1032117;
        double r1032119 = r1032118 + r1032115;
        double r1032120 = r1032119 / r1032117;
        double r1032121 = r1032115 / r1032120;
        double r1032122 = r1032116 / r1032115;
        double r1032123 = r1032121 / r1032122;
        double r1032124 = r1032117 / r1032122;
        double r1032125 = r1032118 - r1032115;
        double r1032126 = r1032124 / r1032125;
        double r1032127 = r1032123 * r1032126;
        return r1032127;
}

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-*-un-lft-identity0.9

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

  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-*-un-lft-identity0.4

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

  12. Applied associate-/l*0.5

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

  14. Applied associate-/r/0.5

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

  15. Applied associate-/r*0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019155 
(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))))