Average Error: 47.1 → 0.2
Time: 1.6s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{i}{16 \cdot i - 4 \cdot \frac{1}{i}}\]
\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{i}{16 \cdot i - 4 \cdot \frac{1}{i}}
double f(double i) {
        double r106628 = i;
        double r106629 = r106628 * r106628;
        double r106630 = r106629 * r106629;
        double r106631 = 2.0;
        double r106632 = r106631 * r106628;
        double r106633 = r106632 * r106632;
        double r106634 = r106630 / r106633;
        double r106635 = 1.0;
        double r106636 = r106633 - r106635;
        double r106637 = r106634 / r106636;
        return r106637;
}
↓
double f(double i) {
        double r106638 = i;
        double r106639 = 16.0;
        double r106640 = r106639 * r106638;
        double r106641 = 4.0;
        double r106642 = 1.0;
        double r106643 = r106642 / r106638;
        double r106644 = r106641 * r106643;
        double r106645 = r106640 - r106644;
        double r106646 = r106638 / r106645;
        return r106646;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.1

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

  2. Simplified16.4

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

  4. Applied associate-/l*15.9

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

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{i}{\color{blue}{16 \cdot i - 4 \cdot \frac{1}{i}}}\]

  6. Final simplification0.2

    \[\leadsto \frac{i}{16 \cdot i - 4 \cdot \frac{1}{i}}\]

Reproduce

herbie shell --seed 2020047 
(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)))