Average Error: 47.0 → 0.1
Time: 3.2s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2 \cdot i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}}{2 \cdot i + \sqrt{1}}\]
\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 i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}}{2 \cdot i + \sqrt{1}}
double f(double i) {
        double r82265 = i;
        double r82266 = r82265 * r82265;
        double r82267 = r82266 * r82266;
        double r82268 = 2.0;
        double r82269 = r82268 * r82265;
        double r82270 = r82269 * r82269;
        double r82271 = r82267 / r82270;
        double r82272 = 1.0;
        double r82273 = r82270 - r82272;
        double r82274 = r82271 / r82273;
        return r82274;
}
↓
double f(double i) {
        double r82275 = i;
        double r82276 = 2.0;
        double r82277 = r82276 * r82275;
        double r82278 = 1.0;
        double r82279 = sqrt(r82278);
        double r82280 = r82277 - r82279;
        double r82281 = r82275 / r82280;
        double r82282 = r82276 * r82276;
        double r82283 = r82275 / r82282;
        double r82284 = r82281 * r82283;
        double r82285 = r82277 + r82279;
        double r82286 = r82284 / r82285;
        return r82286;
}

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. Simplified16.2

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

  4. Applied times-frac15.8

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

  6. Applied add-sqr-sqrt15.8

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

  7. Applied difference-of-squares15.8

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

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

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

  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-*l/0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 
(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)))