Average Error: 47.0 → 0.1
Time: 11.5s
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 r57646 = i;
        double r57647 = r57646 * r57646;
        double r57648 = r57647 * r57647;
        double r57649 = 2.0;
        double r57650 = r57649 * r57646;
        double r57651 = r57650 * r57650;
        double r57652 = r57648 / r57651;
        double r57653 = 1.0;
        double r57654 = r57651 - r57653;
        double r57655 = r57652 / r57654;
        return r57655;
}
↓
double f(double i) {
        double r57656 = 1.0;
        double r57657 = 2.0;
        double r57658 = r57656 / r57657;
        double r57659 = 1.0;
        double r57660 = sqrt(r57659);
        double r57661 = i;
        double r57662 = r57660 / r57661;
        double r57663 = r57657 + r57662;
        double r57664 = r57658 / r57663;
        double r57665 = r57657 - r57662;
        double r57666 = r57658 / r57665;
        double r57667 = r57664 * r57666;
        return r57667;
}

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 2019212 
(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)))