Average Error: 2.4 → 0.4
Time: 26.0s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{1.0}{\frac{2 \cdot i - 1.0}{i} \cdot \frac{2}{1.0}}\]
\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{i}{\frac{2}{1.0}}}{2 \cdot i + 1.0} \cdot \frac{1.0}{\frac{2 \cdot i - 1.0}{i} \cdot \frac{2}{1.0}}
double f(double i) {
        double r1070634 = i;
        double r1070635 = r1070634 * r1070634;
        double r1070636 = r1070635 * r1070635;
        double r1070637 = 2.0;
        double r1070638 = /* ERROR: no posit support in C */;
        double r1070639 = r1070638 * r1070634;
        double r1070640 = r1070639 * r1070639;
        double r1070641 = r1070636 / r1070640;
        double r1070642 = 1.0;
        double r1070643 = /* ERROR: no posit support in C */;
        double r1070644 = r1070640 - r1070643;
        double r1070645 = r1070641 / r1070644;
        return r1070645;
}
↓
double f(double i) {
        double r1070646 = i;
        double r1070647 = 2.0;
        double r1070648 = 1.0;
        double r1070649 = r1070647 / r1070648;
        double r1070650 = r1070646 / r1070649;
        double r1070651 = r1070647 * r1070646;
        double r1070652 = r1070651 + r1070648;
        double r1070653 = r1070650 / r1070652;
        double r1070654 = r1070651 - r1070648;
        double r1070655 = r1070654 / r1070646;
        double r1070656 = r1070655 * r1070649;
        double r1070657 = r1070648 / r1070656;
        double r1070658 = r1070653 * r1070657;
        return r1070658;
}

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

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

  12. Applied associate-/l*0.4

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

  14. Applied associate-/r/0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019154 
(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))))