Average Error: 2.3 → 0.4
Time: 22.5s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}\]
\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{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}
double f(double i) {
        double r1740053 = i;
        double r1740054 = r1740053 * r1740053;
        double r1740055 = r1740054 * r1740054;
        double r1740056 = 2.0;
        double r1740057 = /* ERROR: no posit support in C */;
        double r1740058 = r1740057 * r1740053;
        double r1740059 = r1740058 * r1740058;
        double r1740060 = r1740055 / r1740059;
        double r1740061 = 1.0;
        double r1740062 = /* ERROR: no posit support in C */;
        double r1740063 = r1740059 - r1740062;
        double r1740064 = r1740060 / r1740063;
        return r1740064;
}
↓
double f(double i) {
        double r1740065 = i;
        double r1740066 = 2.0;
        double r1740067 = r1740065 / r1740066;
        double r1740068 = r1740065 * r1740066;
        double r1740069 = 1.0;
        double r1740070 = r1740068 - r1740069;
        double r1740071 = r1740067 / r1740070;
        double r1740072 = r1740067 * r1740071;
        double r1740073 = r1740068 + r1740069;
        double r1740074 = r1740072 / r1740073;
        return r1740074;
}

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. Simplified2.3

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

  4. Applied p16-times-frac1.2

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

  6. Applied associate-*r/1.2

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

  7. Applied associate-*l/1.1

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

  8. Simplified0.9

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

  10. Applied difference-of-sqr-10.8

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

  11. Applied p16-times-frac0.4

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

  13. Applied associate-*l/0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 
(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))))