Average Error: 47.1 → 0.2
Time: 4.4s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\frac{i}{16 \cdot i - \frac{4}{i}}\]
\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 - \frac{4}{i}}
double f(double i) {
        double r50630 = i;
        double r50631 = r50630 * r50630;
        double r50632 = r50631 * r50631;
        double r50633 = 2.0;
        double r50634 = r50633 * r50630;
        double r50635 = r50634 * r50634;
        double r50636 = r50632 / r50635;
        double r50637 = 1.0;
        double r50638 = r50635 - r50637;
        double r50639 = r50636 / r50638;
        return r50639;
}
↓
double f(double i) {
        double r50640 = i;
        double r50641 = 16.0;
        double r50642 = r50641 * r50640;
        double r50643 = 4.0;
        double r50644 = r50643 / r50640;
        double r50645 = r50642 - r50644;
        double r50646 = r50640 / r50645;
        return r50646;
}

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. Simplified16.0

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))