Average Error: 46.1 → 0.2
Time: 10.1s
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 r53887 = i;
        double r53888 = r53887 * r53887;
        double r53889 = r53888 * r53888;
        double r53890 = 2.0;
        double r53891 = r53890 * r53887;
        double r53892 = r53891 * r53891;
        double r53893 = r53889 / r53892;
        double r53894 = 1.0;
        double r53895 = r53892 - r53894;
        double r53896 = r53893 / r53895;
        return r53896;
}
↓
double f(double i) {
        double r53897 = i;
        double r53898 = 16.0;
        double r53899 = r53898 * r53897;
        double r53900 = 4.0;
        double r53901 = 1.0;
        double r53902 = r53901 / r53897;
        double r53903 = r53900 * r53902;
        double r53904 = r53899 - r53903;
        double r53905 = r53897 / r53904;
        return r53905;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.1

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

  2. Simplified15.8

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

  4. Applied associate-/l*15.4

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

  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 2019298 
(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)))