Average Error: 46.8 → 0.5
Time: 4.4s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\sqrt{i}}{2 \cdot i + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{2 \cdot i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}\right)\]
\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{\sqrt{i}}{2 \cdot i + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{2 \cdot i - \sqrt{1}} \cdot \frac{i}{2 \cdot 2}\right)
double f(double i) {
        double r102577 = i;
        double r102578 = r102577 * r102577;
        double r102579 = r102578 * r102578;
        double r102580 = 2.0;
        double r102581 = r102580 * r102577;
        double r102582 = r102581 * r102581;
        double r102583 = r102579 / r102582;
        double r102584 = 1.0;
        double r102585 = r102582 - r102584;
        double r102586 = r102583 / r102585;
        return r102586;
}
↓
double f(double i) {
        double r102587 = i;
        double r102588 = sqrt(r102587);
        double r102589 = 2.0;
        double r102590 = r102589 * r102587;
        double r102591 = 1.0;
        double r102592 = sqrt(r102591);
        double r102593 = r102590 + r102592;
        double r102594 = r102588 / r102593;
        double r102595 = r102590 - r102592;
        double r102596 = r102588 / r102595;
        double r102597 = r102589 * r102589;
        double r102598 = r102587 / r102597;
        double r102599 = r102596 * r102598;
        double r102600 = r102594 * r102599;
        return r102600;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.8

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

  2. Simplified16.5

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

  4. Applied times-frac16.1

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

  6. Applied add-sqr-sqrt16.1

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

  7. Applied difference-of-squares16.1

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

  8. Applied add-sqr-sqrt16.3

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

  9. Applied times-frac0.5

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

  10. Applied associate-*l*0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019318 
(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)))