Average Error: 45.4 → 0.1
Time: 16.9s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\sqrt{i}}{2 \cdot \sqrt{i} - \sqrt{\frac{1.0}{i}}} \cdot \frac{\sqrt{i}}{4 \cdot \left(\sqrt{\frac{1.0}{i}} + 2 \cdot \sqrt{i}\right)}\]
\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{\sqrt{i}}{2 \cdot \sqrt{i} - \sqrt{\frac{1.0}{i}}} \cdot \frac{\sqrt{i}}{4 \cdot \left(\sqrt{\frac{1.0}{i}} + 2 \cdot \sqrt{i}\right)}
double f(double i) {
        double r5662535 = i;
        double r5662536 = r5662535 * r5662535;
        double r5662537 = r5662536 * r5662536;
        double r5662538 = 2.0;
        double r5662539 = r5662538 * r5662535;
        double r5662540 = r5662539 * r5662539;
        double r5662541 = r5662537 / r5662540;
        double r5662542 = 1.0;
        double r5662543 = r5662540 - r5662542;
        double r5662544 = r5662541 / r5662543;
        return r5662544;
}
↓
double f(double i) {
        double r5662545 = i;
        double r5662546 = sqrt(r5662545);
        double r5662547 = 2.0;
        double r5662548 = r5662547 * r5662546;
        double r5662549 = 1.0;
        double r5662550 = r5662549 / r5662545;
        double r5662551 = sqrt(r5662550);
        double r5662552 = r5662548 - r5662551;
        double r5662553 = r5662546 / r5662552;
        double r5662554 = 4.0;
        double r5662555 = r5662551 + r5662548;
        double r5662556 = r5662554 * r5662555;
        double r5662557 = r5662546 / r5662556;
        double r5662558 = r5662553 * r5662557;
        return r5662558;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 45.4

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{i}{4 \cdot \left(4 \cdot i - \frac{1.0}{i}\right)}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied unswap-sqr0.5

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

  8. Applied difference-of-squares0.5

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

  9. Applied associate-*r*0.5

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

  10. Applied add-sqr-sqrt0.3

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

  11. Applied times-frac0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 
(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)))