Average Error: 47.0 → 0.1
Time: 9.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}\]
\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{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}
double f(double i) {
        double r71847 = i;
        double r71848 = r71847 * r71847;
        double r71849 = r71848 * r71848;
        double r71850 = 2.0;
        double r71851 = r71850 * r71847;
        double r71852 = r71851 * r71851;
        double r71853 = r71849 / r71852;
        double r71854 = 1.0;
        double r71855 = r71852 - r71854;
        double r71856 = r71853 / r71855;
        return r71856;
}
↓
double f(double i) {
        double r71857 = 1.0;
        double r71858 = 2.0;
        double r71859 = r71857 / r71858;
        double r71860 = 1.0;
        double r71861 = sqrt(r71860);
        double r71862 = i;
        double r71863 = r71861 / r71862;
        double r71864 = r71858 + r71863;
        double r71865 = r71859 / r71864;
        double r71866 = r71858 - r71863;
        double r71867 = r71859 / r71866;
        double r71868 = r71865 * r71867;
        return r71868;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.0

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

  2. Simplified0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied times-frac0.4

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

  6. Applied difference-of-squares0.4

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

  7. Applied add-sqr-sqrt1.4

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

  8. Applied add-sqr-sqrt1.9

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

  9. Applied add-cube-cbrt1.9

    \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{2} \cdot \sqrt{2}}}{\sqrt{2} \cdot \sqrt{2}}}{\left(2 + \frac{\sqrt{1}}{i}\right) \cdot \left(2 - \frac{\sqrt{1}}{i}\right)}\]

  10. Applied times-frac1.9

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{2}} \cdot \frac{\sqrt[3]{1}}{\sqrt{2}}}}{\sqrt{2} \cdot \sqrt{2}}}{\left(2 + \frac{\sqrt{1}}{i}\right) \cdot \left(2 - \frac{\sqrt{1}}{i}\right)}\]

  11. Applied times-frac1.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{2}}}{\sqrt{2}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{2}}}{\sqrt{2}}}}{\left(2 + \frac{\sqrt{1}}{i}\right) \cdot \left(2 - \frac{\sqrt{1}}{i}\right)}\]

  12. Applied times-frac1.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{2}}}{\sqrt{2}}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{\sqrt{2}}}{\sqrt{2}}}{2 - \frac{\sqrt{1}}{i}}}\]

  13. Simplified0.7

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}}} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{\sqrt{2}}}{\sqrt{2}}}{2 - \frac{\sqrt{1}}{i}}\]

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))