Average Error: 2.3 → 0.5
Time: 56.8s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{i \cdot \frac{\frac{i}{2 \cdot 2}}{i \cdot 2 + 1.0}}{i \cdot 2 - 1.0}\]
\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{i \cdot \frac{\frac{i}{2 \cdot 2}}{i \cdot 2 + 1.0}}{i \cdot 2 - 1.0}
double f(double i) {
        double r1709083 = i;
        double r1709084 = r1709083 * r1709083;
        double r1709085 = r1709084 * r1709084;
        double r1709086 = 2.0;
        double r1709087 = /* ERROR: no posit support in C */;
        double r1709088 = r1709087 * r1709083;
        double r1709089 = r1709088 * r1709088;
        double r1709090 = r1709085 / r1709089;
        double r1709091 = 1.0;
        double r1709092 = /* ERROR: no posit support in C */;
        double r1709093 = r1709089 - r1709092;
        double r1709094 = r1709090 / r1709093;
        return r1709094;
}
↓
double f(double i) {
        double r1709095 = i;
        double r1709096 = 2.0;
        double r1709097 = r1709096 * r1709096;
        double r1709098 = r1709095 / r1709097;
        double r1709099 = r1709095 * r1709096;
        double r1709100 = 1.0;
        double r1709101 = r1709099 + r1709100;
        double r1709102 = r1709098 / r1709101;
        double r1709103 = r1709095 * r1709102;
        double r1709104 = r1709099 - r1709100;
        double r1709105 = r1709103 / r1709104;
        return r1709105;
}

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. Simplified1.1

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

  4. Applied p16-*-un-lft-identity1.1

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

  5. Applied difference-of-squares1.1

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

  6. Applied associate-*r*1.1

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

  7. Applied associate-*r*1.1

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

  8. Applied associate-/r*0.7

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

  9. Applied associate-*r/0.6

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

  11. Applied associate-*r*0.6

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

  12. Applied associate-/r*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019158 
(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))))