Average Error: 46.3 → 0.1
Time: 15.1s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{2 + \frac{\sqrt{1}}{i}}}{2} \cdot \frac{\frac{1}{2 - \frac{\sqrt{1}}{i}}}{2}\]
\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}{2 + \frac{\sqrt{1}}{i}}}{2} \cdot \frac{\frac{1}{2 - \frac{\sqrt{1}}{i}}}{2}
double f(double i) {
        double r81061 = i;
        double r81062 = r81061 * r81061;
        double r81063 = r81062 * r81062;
        double r81064 = 2.0;
        double r81065 = r81064 * r81061;
        double r81066 = r81065 * r81065;
        double r81067 = r81063 / r81066;
        double r81068 = 1.0;
        double r81069 = r81066 - r81068;
        double r81070 = r81067 / r81069;
        return r81070;
}
↓
double f(double i) {
        double r81071 = 1.0;
        double r81072 = 2.0;
        double r81073 = 1.0;
        double r81074 = sqrt(r81073);
        double r81075 = i;
        double r81076 = r81074 / r81075;
        double r81077 = r81072 + r81076;
        double r81078 = r81071 / r81077;
        double r81079 = r81078 / r81072;
        double r81080 = r81072 - r81076;
        double r81081 = r81071 / r81080;
        double r81082 = r81081 / r81072;
        double r81083 = r81079 * r81082;
        return r81083;
}

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. Simplified0.3

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

  4. Applied div-inv0.3

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

  5. Simplified0.3

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

  7. Applied add-sqr-sqrt0.3

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

  8. Applied times-frac0.4

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

  9. Applied difference-of-squares0.4

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

  10. Applied add-cube-cbrt0.4

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

  11. Applied times-frac0.1

    \[\leadsto 1 \cdot \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\sqrt[3]{1}}{2 - \frac{\sqrt{1}}{i}}}}{2 \cdot 2}\]

  12. Applied times-frac0.1

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

  13. Simplified0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019351 +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)))