Average Error: 46.7 → 0.2
Time: 15.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2}}{i \cdot 2 - \sqrt{1}} \cdot \left(\sqrt{\frac{\frac{i}{2}}{i \cdot 2 + \sqrt{1}}} \cdot \sqrt{\frac{\frac{i}{2}}{i \cdot 2 + \sqrt{1}}}\right)\]
\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{\frac{i}{2}}{i \cdot 2 - \sqrt{1}} \cdot \left(\sqrt{\frac{\frac{i}{2}}{i \cdot 2 + \sqrt{1}}} \cdot \sqrt{\frac{\frac{i}{2}}{i \cdot 2 + \sqrt{1}}}\right)
double f(double i) {
        double r2768237 = i;
        double r2768238 = r2768237 * r2768237;
        double r2768239 = r2768238 * r2768238;
        double r2768240 = 2.0;
        double r2768241 = r2768240 * r2768237;
        double r2768242 = r2768241 * r2768241;
        double r2768243 = r2768239 / r2768242;
        double r2768244 = 1.0;
        double r2768245 = r2768242 - r2768244;
        double r2768246 = r2768243 / r2768245;
        return r2768246;
}
↓
double f(double i) {
        double r2768247 = i;
        double r2768248 = 2.0;
        double r2768249 = r2768247 / r2768248;
        double r2768250 = r2768247 * r2768248;
        double r2768251 = 1.0;
        double r2768252 = sqrt(r2768251);
        double r2768253 = r2768250 - r2768252;
        double r2768254 = r2768249 / r2768253;
        double r2768255 = r2768250 + r2768252;
        double r2768256 = r2768249 / r2768255;
        double r2768257 = sqrt(r2768256);
        double r2768258 = r2768257 * r2768257;
        double r2768259 = r2768254 * r2768258;
        return r2768259;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.7

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

  2. Simplified15.6

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

  4. Applied add-sqr-sqrt15.6

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

  5. Applied difference-of-squares15.6

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

  6. Applied times-frac0.0

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

  8. Applied add-sqr-sqrt0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))