Average Error: 46.6 → 0.0
Time: 3.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2 \cdot 2} \cdot \frac{i}{2 \cdot i + \sqrt{1}}}{2 \cdot i - \sqrt{1}}\]
\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{i}{2 \cdot 2} \cdot \frac{i}{2 \cdot i + \sqrt{1}}}{2 \cdot i - \sqrt{1}}
double f(double i) {
        double r85721 = i;
        double r85722 = r85721 * r85721;
        double r85723 = r85722 * r85722;
        double r85724 = 2.0;
        double r85725 = r85724 * r85721;
        double r85726 = r85725 * r85725;
        double r85727 = r85723 / r85726;
        double r85728 = 1.0;
        double r85729 = r85726 - r85728;
        double r85730 = r85727 / r85729;
        return r85730;
}
↓
double f(double i) {
        double r85731 = i;
        double r85732 = 2.0;
        double r85733 = r85732 * r85732;
        double r85734 = r85731 / r85733;
        double r85735 = r85732 * r85731;
        double r85736 = 1.0;
        double r85737 = sqrt(r85736);
        double r85738 = r85735 + r85737;
        double r85739 = r85731 / r85738;
        double r85740 = r85734 * r85739;
        double r85741 = r85735 - r85737;
        double r85742 = r85740 / r85741;
        return r85742;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.6

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

  2. Simplified16.0

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

  4. Applied times-frac15.6

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

  6. Applied add-sqr-sqrt15.6

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

  7. Applied difference-of-squares15.6

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

  8. Applied *-un-lft-identity15.6

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

  9. Applied times-frac0.1

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

  11. Applied associate-*r/0.1

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

  12. Applied associate-*l/0.1

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020081 
(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)))