Average Error: 46.2 → 0.1
Time: 3.4s
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 r103229 = i;
        double r103230 = r103229 * r103229;
        double r103231 = r103230 * r103230;
        double r103232 = 2.0;
        double r103233 = r103232 * r103229;
        double r103234 = r103233 * r103233;
        double r103235 = r103231 / r103234;
        double r103236 = 1.0;
        double r103237 = r103234 - r103236;
        double r103238 = r103235 / r103237;
        return r103238;
}
↓
double f(double i) {
        double r103239 = i;
        double r103240 = 2.0;
        double r103241 = r103240 * r103240;
        double r103242 = r103239 / r103241;
        double r103243 = r103240 * r103239;
        double r103244 = 1.0;
        double r103245 = sqrt(r103244);
        double r103246 = r103243 + r103245;
        double r103247 = r103239 / r103246;
        double r103248 = r103242 * r103247;
        double r103249 = r103243 - r103245;
        double r103250 = r103248 / r103249;
        return r103250;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.2

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

  2. Simplified16.4

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

  4. Applied times-frac16.0

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

  6. Applied add-sqr-sqrt16.0

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

  7. Applied difference-of-squares16.0

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

  8. Applied *-un-lft-identity16.0

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

  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-*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/0.2

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

  13. Simplified0.1

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

  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 2020100 
(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)))