Average Error: 46.3 → 0.1
Time: 40.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{1}{\frac{\sqrt{1}}{i} + 2} \cdot \frac{\frac{i}{2}}{\frac{2 \cdot i - \sqrt{1}}{\frac{1}{2}}}\]
\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}
↓
\frac{1}{\frac{\sqrt{1}}{i} + 2} \cdot \frac{\frac{i}{2}}{\frac{2 \cdot i - \sqrt{1}}{\frac{1}{2}}}
double f(double i) {
        double r6205850 = i;
        double r6205851 = r6205850 * r6205850;
        double r6205852 = r6205851 * r6205851;
        double r6205853 = 2.0;
        double r6205854 = r6205853 * r6205850;
        double r6205855 = r6205854 * r6205854;
        double r6205856 = r6205852 / r6205855;
        double r6205857 = 1.0;
        double r6205858 = r6205855 - r6205857;
        double r6205859 = r6205856 / r6205858;
        return r6205859;
}
↓
double f(double i) {
        double r6205860 = 1.0;
        double r6205861 = 1.0;
        double r6205862 = sqrt(r6205861);
        double r6205863 = i;
        double r6205864 = r6205862 / r6205863;
        double r6205865 = 2.0;
        double r6205866 = r6205864 + r6205865;
        double r6205867 = r6205860 / r6205866;
        double r6205868 = r6205863 / r6205865;
        double r6205869 = r6205865 * r6205863;
        double r6205870 = r6205869 - r6205862;
        double r6205871 = r6205860 / r6205865;
        double r6205872 = r6205870 / r6205871;
        double r6205873 = r6205868 / r6205872;
        double r6205874 = r6205867 * r6205873;
        return r6205874;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.3

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

  2. Simplified15.2

    \[\leadsto \color{blue}{\frac{\frac{i}{2}}{\frac{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}{\frac{i}{2}}}}\]
  3. Using strategy rm

  4. Applied div-inv15.2

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

  5. Applied add-sqr-sqrt15.2

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

  6. Applied difference-of-squares15.2

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

  7. Applied times-frac0.1

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

  8. Applied *-un-lft-identity0.1

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

  9. Applied *-un-lft-identity0.1

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

  10. Applied times-frac0.1

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

  11. Applied times-frac0.1

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

  12. Simplified0.1

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

  13. Taylor expanded around inf 0.1

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

  14. Final simplification0.1

    \[\leadsto \frac{1}{\frac{\sqrt{1}}{i} + 2} \cdot \frac{\frac{i}{2}}{\frac{2 \cdot i - \sqrt{1}}{\frac{1}{2}}}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))