Average Error: 46.3 → 0.2
Time: 12.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{\left(2 \cdot 2\right) \cdot i - \frac{1}{i}}}{2} \cdot \frac{i}{2}\]
\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}
↓
\frac{\frac{1}{\left(2 \cdot 2\right) \cdot i - \frac{1}{i}}}{2} \cdot \frac{i}{2}
double f(double i) {
        double r84692 = i;
        double r84693 = r84692 * r84692;
        double r84694 = r84693 * r84693;
        double r84695 = 2.0;
        double r84696 = r84695 * r84692;
        double r84697 = r84696 * r84696;
        double r84698 = r84694 / r84697;
        double r84699 = 1.0;
        double r84700 = r84697 - r84699;
        double r84701 = r84698 / r84700;
        return r84701;
}
↓
double f(double i) {
        double r84702 = 1.0;
        double r84703 = 2.0;
        double r84704 = r84703 * r84703;
        double r84705 = i;
        double r84706 = r84704 * r84705;
        double r84707 = 1.0;
        double r84708 = r84707 / r84705;
        double r84709 = r84706 - r84708;
        double r84710 = r84702 / r84709;
        double r84711 = r84710 / r84703;
        double r84712 = r84705 / r84703;
        double r84713 = r84711 * r84712;
        return r84713;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.3

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

  2. Simplified0.2

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

  4. Applied associate-/r*0.1

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

  6. Applied clear-num0.5

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

  8. Applied div-inv0.6

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

  9. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\left(2 \cdot 2\right) \cdot i - \frac{1}{i}\right) \cdot \frac{1}{i}}}{2 \cdot 2}\]

  10. Applied times-frac0.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(2 \cdot 2\right) \cdot i - \frac{1}{i}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{i}}}}{2 \cdot 2}\]

  11. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left(2 \cdot 2\right) \cdot i - \frac{1}{i}}}{2} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{1}{i}}}{2}}\]

  12. Simplified0.2

    \[\leadsto \color{blue}{\frac{\frac{1}{\left(2 \cdot 2\right) \cdot i - \frac{1}{i}}}{2}} \cdot \frac{\frac{\sqrt[3]{1}}{\frac{1}{i}}}{2}\]

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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)))