Average Error: 46.3 → 0.1
Time: 16.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{i}{2 \cdot i + \sqrt{1}} \cdot \frac{1}{\frac{\left(2 \cdot i - \sqrt{1}\right) \cdot 2}{\frac{i}{2}}}\]
\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{i}{2 \cdot i + \sqrt{1}} \cdot \frac{1}{\frac{\left(2 \cdot i - \sqrt{1}\right) \cdot 2}{\frac{i}{2}}}
double f(double i) {
        double r6118793 = i;
        double r6118794 = r6118793 * r6118793;
        double r6118795 = r6118794 * r6118794;
        double r6118796 = 2.0;
        double r6118797 = r6118796 * r6118793;
        double r6118798 = r6118797 * r6118797;
        double r6118799 = r6118795 / r6118798;
        double r6118800 = 1.0;
        double r6118801 = r6118798 - r6118800;
        double r6118802 = r6118799 / r6118801;
        return r6118802;
}
↓
double f(double i) {
        double r6118803 = i;
        double r6118804 = 2.0;
        double r6118805 = r6118804 * r6118803;
        double r6118806 = 1.0;
        double r6118807 = sqrt(r6118806);
        double r6118808 = r6118805 + r6118807;
        double r6118809 = r6118803 / r6118808;
        double r6118810 = 1.0;
        double r6118811 = r6118805 - r6118807;
        double r6118812 = r6118811 * r6118804;
        double r6118813 = r6118803 / r6118804;
        double r6118814 = r6118812 / r6118813;
        double r6118815 = r6118810 / r6118814;
        double r6118816 = r6118809 * r6118815;
        return r6118816;
}

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

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

  4. Applied add-sqr-sqrt15.2

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

  5. Applied difference-of-squares15.2

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

  6. Applied div-inv15.2

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

  7. Applied times-frac0.1

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

  8. Applied associate-*l*0.1

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

  9. Simplified0.1

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

  11. Applied clear-num0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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)))