Average Error: 2.4 → 0.5
Time: 24.3s
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 r2621065 = i;
        double r2621066 = r2621065 * r2621065;
        double r2621067 = r2621066 * r2621066;
        double r2621068 = 2.0;
        double r2621069 = /* ERROR: no posit support in C */;
        double r2621070 = r2621069 * r2621065;
        double r2621071 = r2621070 * r2621070;
        double r2621072 = r2621067 / r2621071;
        double r2621073 = 1.0;
        double r2621074 = /* ERROR: no posit support in C */;
        double r2621075 = r2621071 - r2621074;
        double r2621076 = r2621072 / r2621075;
        return r2621076;
}
↓
double f(double i) {
        double r2621077 = i;
        double r2621078 = 2.0;
        double r2621079 = r2621077 / r2621078;
        double r2621080 = r2621077 * r2621078;
        double r2621081 = 1.0;
        double r2621082 = r2621080 + r2621081;
        double r2621083 = r2621079 / r2621082;
        double r2621084 = r2621077 * r2621083;
        double r2621085 = r2621084 / r2621078;
        double r2621086 = r2621080 - r2621081;
        double r2621087 = r2621085 / r2621086;
        return r2621087;
}

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 
(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))))