Average Error: 2.3 → 0.5
Time: 51.3s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i \cdot \frac{\frac{i}{2}}{i \cdot 2 + 1.0}}{2}}{i \cdot 2 - 1.0}\]
\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 r4169397 = i;
        double r4169398 = r4169397 * r4169397;
        double r4169399 = r4169398 * r4169398;
        double r4169400 = 2.0;
        double r4169401 = /* ERROR: no posit support in C */;
        double r4169402 = r4169401 * r4169397;
        double r4169403 = r4169402 * r4169402;
        double r4169404 = r4169399 / r4169403;
        double r4169405 = 1.0;
        double r4169406 = /* ERROR: no posit support in C */;
        double r4169407 = r4169403 - r4169406;
        double r4169408 = r4169404 / r4169407;
        return r4169408;
}
↓
double f(double i) {
        double r4169409 = i;
        double r4169410 = 2.0;
        double r4169411 = r4169409 / r4169410;
        double r4169412 = r4169409 * r4169410;
        double r4169413 = 1.0;
        double r4169414 = r4169412 + r4169413;
        double r4169415 = r4169411 / r4169414;
        double r4169416 = r4169409 * r4169415;
        double r4169417 = r4169416 / r4169410;
        double r4169418 = r4169412 - r4169413;
        double r4169419 = r4169417 / r4169418;
        return r4169419;
}

Error

Bits error versus i

Derivation

  1. Initial program 2.3

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

  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*0.5

    \[\leadsto \left(\frac{i}{\left(2\right)}\right) \cdot \color{blue}{\left(\frac{\left(\frac{\left(\frac{i}{\left(2\right)}\right)}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(1.0\right)}\right)}\right)}{\left(\left(i \cdot \left(2\right)\right) - \left(1.0\right)\right)}\right)}\]
  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 2019133 
(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))))