Average Error: 45.8 → 0.1
Time: 17.2s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)\]
\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}
↓
\frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)
double f(double i) {
        double r1733823 = i;
        double r1733824 = r1733823 * r1733823;
        double r1733825 = r1733824 * r1733824;
        double r1733826 = 2.0;
        double r1733827 = r1733826 * r1733823;
        double r1733828 = r1733827 * r1733827;
        double r1733829 = r1733825 / r1733828;
        double r1733830 = 1.0;
        double r1733831 = r1733828 - r1733830;
        double r1733832 = r1733829 / r1733831;
        return r1733832;
}
↓
double f(double i) {
        double r1733833 = 0.5;
        double r1733834 = 2.0;
        double r1733835 = 1.0;
        double r1733836 = sqrt(r1733835);
        double r1733837 = i;
        double r1733838 = r1733836 / r1733837;
        double r1733839 = r1733834 - r1733838;
        double r1733840 = r1733833 / r1733839;
        double r1733841 = 1.0;
        double r1733842 = r1733834 + r1733838;
        double r1733843 = r1733841 / r1733842;
        double r1733844 = r1733840 * r1733843;
        double r1733845 = r1733833 * r1733844;
        return r1733845;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.8

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{4 - \frac{1.0}{i \cdot i}} \cdot \frac{1}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\frac{1}{2}}{4 - \frac{\color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{i \cdot i}} \cdot \frac{1}{2}\]

  5. Applied times-frac0.5

    \[\leadsto \frac{\frac{1}{2}}{4 - \color{blue}{\frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}}} \cdot \frac{1}{2}\]

  6. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{4} \cdot \sqrt{4}} - \frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2}\]

  7. Applied difference-of-squares0.5

    \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)}} \cdot \frac{1}{2}\]

  8. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{2}}}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)} \cdot \frac{1}{2}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{4} + \frac{\sqrt{1.0}}{i}} \cdot \frac{\frac{1}{2}}{\sqrt{4} - \frac{\sqrt{1.0}}{i}}\right)} \cdot \frac{1}{2}\]

  10. Final simplification0.1

    \[\leadsto \frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))