Average Error: 47.0 → 0.1
Time: 3.8s
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 r109150 = i;
        double r109151 = r109150 * r109150;
        double r109152 = r109151 * r109151;
        double r109153 = 2.0;
        double r109154 = r109153 * r109150;
        double r109155 = r109154 * r109154;
        double r109156 = r109152 / r109155;
        double r109157 = 1.0;
        double r109158 = r109155 - r109157;
        double r109159 = r109156 / r109158;
        return r109159;
}
↓
double f(double i) {
        double r109160 = i;
        double r109161 = 2.0;
        double r109162 = r109161 * r109161;
        double r109163 = r109160 / r109162;
        double r109164 = r109161 * r109160;
        double r109165 = 1.0;
        double r109166 = sqrt(r109165);
        double r109167 = r109164 - r109166;
        double r109168 = r109160 / r109167;
        double r109169 = r109163 * r109168;
        double r109170 = r109164 + r109166;
        double r109171 = r109169 / r109170;
        return r109171;
}

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. 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-frac0.2

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

  11. Applied associate-*l/0.2

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

  12. Applied associate-*l/0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020002 
(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)))