Average Error: 2.4 → 0.5
Time: 22.6s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i \cdot \frac{\frac{i}{2}}{i \cdot 2 + 1.0}}{2}}{i \cdot 2 - 1.0}\]
\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 \cdot \frac{\frac{i}{2}}{i \cdot 2 + 1.0}}{2}}{i \cdot 2 - 1.0}
double f(double i) {
        double r1925574 = i;
        double r1925575 = r1925574 * r1925574;
        double r1925576 = r1925575 * r1925575;
        double r1925577 = 2.0;
        double r1925578 = /* ERROR: no posit support in C */;
        double r1925579 = r1925578 * r1925574;
        double r1925580 = r1925579 * r1925579;
        double r1925581 = r1925576 / r1925580;
        double r1925582 = 1.0;
        double r1925583 = /* ERROR: no posit support in C */;
        double r1925584 = r1925580 - r1925583;
        double r1925585 = r1925581 / r1925584;
        return r1925585;
}
↓
double f(double i) {
        double r1925586 = i;
        double r1925587 = 2.0;
        double r1925588 = r1925586 / r1925587;
        double r1925589 = r1925586 * r1925587;
        double r1925590 = 1.0;
        double r1925591 = r1925589 + r1925590;
        double r1925592 = r1925588 / r1925591;
        double r1925593 = r1925586 * r1925592;
        double r1925594 = r1925593 / r1925587;
        double r1925595 = r1925589 - r1925590;
        double r1925596 = r1925594 / r1925595;
        return r1925596;
}

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. Simplified0.9

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

  4. Applied difference-of-sqr-10.8

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

  5. Applied associate-/r*0.6

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

  7. Applied associate-*r/0.5

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

  9. Applied associate-*l/0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019128 +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))))