Average Error: 2.4 → 0.4
Time: 26.9s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{2}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{2}}{2 \cdot i - 1.0}\]
\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{i}{2}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{2}}{2 \cdot i - 1.0}
double f(double i) {
        double r4747605 = i;
        double r4747606 = r4747605 * r4747605;
        double r4747607 = r4747606 * r4747606;
        double r4747608 = 2.0;
        double r4747609 = /* ERROR: no posit support in C */;
        double r4747610 = r4747609 * r4747605;
        double r4747611 = r4747610 * r4747610;
        double r4747612 = r4747607 / r4747611;
        double r4747613 = 1.0;
        double r4747614 = /* ERROR: no posit support in C */;
        double r4747615 = r4747611 - r4747614;
        double r4747616 = r4747612 / r4747615;
        return r4747616;
}
↓
double f(double i) {
        double r4747617 = i;
        double r4747618 = 2.0;
        double r4747619 = r4747617 / r4747618;
        double r4747620 = r4747618 * r4747617;
        double r4747621 = 1.0;
        double r4747622 = r4747620 + r4747621;
        double r4747623 = r4747619 / r4747622;
        double r4747624 = r4747620 - r4747621;
        double r4747625 = r4747619 / r4747624;
        double r4747626 = r4747623 * r4747625;
        return r4747626;
}

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 associate-/l*1.0

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

  4. Simplified0.8

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

  6. Applied difference-of-sqr-10.7

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

  7. Applied p16-times-frac0.8

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

  8. Applied p16-times-frac0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 
(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))))