Average Error: 46.4 → 0.1
Time: 11.7s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2 \cdot i - \sqrt{1}} \cdot i}{\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot 2\right)}\]
\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 i - \sqrt{1}} \cdot i}{\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot 2\right)}
double f(double i) {
        double r61080 = i;
        double r61081 = r61080 * r61080;
        double r61082 = r61081 * r61081;
        double r61083 = 2.0;
        double r61084 = r61083 * r61080;
        double r61085 = r61084 * r61084;
        double r61086 = r61082 / r61085;
        double r61087 = 1.0;
        double r61088 = r61085 - r61087;
        double r61089 = r61086 / r61088;
        return r61089;
}
↓
double f(double i) {
        double r61090 = i;
        double r61091 = 2.0;
        double r61092 = r61091 * r61090;
        double r61093 = 1.0;
        double r61094 = sqrt(r61093);
        double r61095 = r61092 - r61094;
        double r61096 = r61090 / r61095;
        double r61097 = r61096 * r61090;
        double r61098 = r61092 + r61094;
        double r61099 = r61091 * r61091;
        double r61100 = r61098 * r61099;
        double r61101 = r61097 / r61100;
        return r61101;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.4

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

  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-frac15.5

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

  6. Applied add-sqr-sqrt15.5

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

  7. Applied difference-of-squares15.5

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

  8. Applied *-un-lft-identity15.5

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

  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. Applied associate-*l*0.1

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

  12. Applied associate-*r/0.1

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

  13. Applied frac-times0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019297 
(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)))