Average Error: 45.9 → 0.1
Time: 7.3s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)\]
\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) - 1.0}
↓
\frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)
double f(double i) {
        double r1071389 = i;
        double r1071390 = r1071389 * r1071389;
        double r1071391 = r1071390 * r1071390;
        double r1071392 = 2.0;
        double r1071393 = r1071392 * r1071389;
        double r1071394 = r1071393 * r1071393;
        double r1071395 = r1071391 / r1071394;
        double r1071396 = 1.0;
        double r1071397 = r1071394 - r1071396;
        double r1071398 = r1071395 / r1071397;
        return r1071398;
}
↓
double f(double i) {
        double r1071399 = 0.5;
        double r1071400 = 2.0;
        double r1071401 = 1.0;
        double r1071402 = sqrt(r1071401);
        double r1071403 = i;
        double r1071404 = r1071402 / r1071403;
        double r1071405 = r1071400 - r1071404;
        double r1071406 = r1071399 / r1071405;
        double r1071407 = 1.0;
        double r1071408 = r1071400 + r1071404;
        double r1071409 = r1071407 / r1071408;
        double r1071410 = r1071406 * r1071409;
        double r1071411 = r1071399 * r1071410;
        return r1071411;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.9

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{2}}{4 - \frac{1.0}{i \cdot i}} \cdot \frac{1}{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{\frac{1}{2}}{4 - \frac{\color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{i \cdot i}} \cdot \frac{1}{2}\]

  5. Applied times-frac0.4

    \[\leadsto \frac{\frac{1}{2}}{4 - \color{blue}{\frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}}} \cdot \frac{1}{2}\]

  6. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{4} \cdot \sqrt{4}} - \frac{\sqrt{1.0}}{i} \cdot \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2}\]

  7. Applied difference-of-squares0.4

    \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)}} \cdot \frac{1}{2}\]

  8. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{2}}}{\left(\sqrt{4} + \frac{\sqrt{1.0}}{i}\right) \cdot \left(\sqrt{4} - \frac{\sqrt{1.0}}{i}\right)} \cdot \frac{1}{2}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{4} + \frac{\sqrt{1.0}}{i}} \cdot \frac{\frac{1}{2}}{\sqrt{4} - \frac{\sqrt{1.0}}{i}}\right)} \cdot \frac{1}{2}\]

  10. Final simplification0.1

    \[\leadsto \frac{1}{2} \cdot \left(\frac{\frac{1}{2}}{2 - \frac{\sqrt{1.0}}{i}} \cdot \frac{1}{2 + \frac{\sqrt{1.0}}{i}}\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))