Average Error: 47.0 → 0.1
Time: 17.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{1}{2}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}\]
\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}}{2 + \frac{\sqrt{1}}{i}} \cdot \frac{\frac{1}{2}}{2 - \frac{\sqrt{1}}{i}}
double f(double i) {
        double r125986 = i;
        double r125987 = r125986 * r125986;
        double r125988 = r125987 * r125987;
        double r125989 = 2.0;
        double r125990 = r125989 * r125986;
        double r125991 = r125990 * r125990;
        double r125992 = r125988 / r125991;
        double r125993 = 1.0;
        double r125994 = r125991 - r125993;
        double r125995 = r125992 / r125994;
        return r125995;
}
↓
double f(double i) {
        double r125996 = 1.0;
        double r125997 = 2.0;
        double r125998 = r125996 / r125997;
        double r125999 = 1.0;
        double r126000 = sqrt(r125999);
        double r126001 = i;
        double r126002 = r126000 / r126001;
        double r126003 = r125997 + r126002;
        double r126004 = r125998 / r126003;
        double r126005 = r125997 - r126002;
        double r126006 = r125998 / r126005;
        double r126007 = r126004 * r126006;
        return r126007;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.0

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

  2. Simplified0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied times-frac0.4

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

  6. Applied difference-of-squares0.4

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

  7. Applied add-cube-cbrt0.4

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.1

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

  10. Simplified0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 +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)))