Average Error: 46.6 → 0.2
Time: 8.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{i}{\left(4 \cdot i - \frac{1}{i}\right) \cdot \left(2 \cdot 2\right)}\]
\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{i}{\left(4 \cdot i - \frac{1}{i}\right) \cdot \left(2 \cdot 2\right)}
double f(double i) {
        double r72354 = i;
        double r72355 = r72354 * r72354;
        double r72356 = r72355 * r72355;
        double r72357 = 2.0;
        double r72358 = r72357 * r72354;
        double r72359 = r72358 * r72358;
        double r72360 = r72356 / r72359;
        double r72361 = 1.0;
        double r72362 = r72359 - r72361;
        double r72363 = r72360 / r72362;
        return r72363;
}
↓
double f(double i) {
        double r72364 = i;
        double r72365 = 4.0;
        double r72366 = r72365 * r72364;
        double r72367 = 1.0;
        double r72368 = r72367 / r72364;
        double r72369 = r72366 - r72368;
        double r72370 = 2.0;
        double r72371 = r72370 * r72370;
        double r72372 = r72369 * r72371;
        double r72373 = r72364 / r72372;
        return r72373;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.6

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

  2. Simplified0.2

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \frac{i}{\color{blue}{\left(4 \cdot i - 1 \cdot \frac{1}{i}\right)} \cdot \left(2 \cdot 2\right)}\]

  4. Simplified0.2

    \[\leadsto \frac{i}{\color{blue}{\left(4 \cdot i - \frac{1}{i}\right)} \cdot \left(2 \cdot 2\right)}\]

  5. Final simplification0.2

    \[\leadsto \frac{i}{\left(4 \cdot i - \frac{1}{i}\right) \cdot \left(2 \cdot 2\right)}\]

Reproduce

herbie shell --seed 2020046 +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)))