Average Error: 46.6 → 0.1
Time: 15.2s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{\frac{\frac{1}{2}}{2}}{2 + \frac{\sqrt{1}}{i}}}{2 - \frac{\sqrt{1}}{i}}\]
\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{\frac{\frac{1}{2}}{2}}{2 + \frac{\sqrt{1}}{i}}}{2 - \frac{\sqrt{1}}{i}}
double f(double i) {
        double r50789 = i;
        double r50790 = r50789 * r50789;
        double r50791 = r50790 * r50790;
        double r50792 = 2.0;
        double r50793 = r50792 * r50789;
        double r50794 = r50793 * r50793;
        double r50795 = r50791 / r50794;
        double r50796 = 1.0;
        double r50797 = r50794 - r50796;
        double r50798 = r50795 / r50797;
        return r50798;
}
↓
double f(double i) {
        double r50799 = 1.0;
        double r50800 = 2.0;
        double r50801 = r50799 / r50800;
        double r50802 = r50801 / r50800;
        double r50803 = 1.0;
        double r50804 = sqrt(r50803);
        double r50805 = i;
        double r50806 = r50804 / r50805;
        double r50807 = r50800 + r50806;
        double r50808 = r50802 / r50807;
        double r50809 = r50800 - r50806;
        double r50810 = r50808 / r50809;
        return r50810;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied times-frac0.5

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

  6. Applied difference-of-squares0.5

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 
(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)))