Average Error: 46.7 → 0.1
Time: 19.8s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{\frac{i}{2 \cdot 2}}{4 \cdot i - \frac{1}{i}}\]
\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{i}{2 \cdot 2}}{4 \cdot i - \frac{1}{i}}
double f(double i) {
        double r2335508 = i;
        double r2335509 = r2335508 * r2335508;
        double r2335510 = r2335509 * r2335509;
        double r2335511 = 2.0;
        double r2335512 = r2335511 * r2335508;
        double r2335513 = r2335512 * r2335512;
        double r2335514 = r2335510 / r2335513;
        double r2335515 = 1.0;
        double r2335516 = r2335513 - r2335515;
        double r2335517 = r2335514 / r2335516;
        return r2335517;
}
↓
double f(double i) {
        double r2335518 = i;
        double r2335519 = 2.0;
        double r2335520 = r2335519 * r2335519;
        double r2335521 = r2335518 / r2335520;
        double r2335522 = 4.0;
        double r2335523 = r2335522 * r2335518;
        double r2335524 = 1.0;
        double r2335525 = r2335524 / r2335518;
        double r2335526 = r2335523 - r2335525;
        double r2335527 = r2335521 / r2335526;
        return r2335527;
}

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. Simplified0.1

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

  3. Taylor expanded around 0 0.1

    \[\leadsto \frac{\frac{i}{2 \cdot 2}}{\color{blue}{4 \cdot i - 1 \cdot \frac{1}{i}}}\]

  4. Simplified0.1

    \[\leadsto \frac{\frac{i}{2 \cdot 2}}{\color{blue}{4 \cdot i - \frac{1}{i}}}\]

  5. Final simplification0.1

    \[\leadsto \frac{\frac{i}{2 \cdot 2}}{4 \cdot i - \frac{1}{i}}\]

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)))