Average Error: 46.1 → 0.0
Time: 23.2s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 219.36859705155612:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 219.36859705155612:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\

\end{array}
double f(double i) {
        double r8583985 = i;
        double r8583986 = r8583985 * r8583985;
        double r8583987 = r8583986 * r8583986;
        double r8583988 = 2.0;
        double r8583989 = r8583988 * r8583985;
        double r8583990 = r8583989 * r8583989;
        double r8583991 = r8583987 / r8583990;
        double r8583992 = 1.0;
        double r8583993 = r8583990 - r8583992;
        double r8583994 = r8583991 / r8583993;
        return r8583994;
}
↓
double f(double i) {
        double r8583995 = i;
        double r8583996 = 219.36859705155612;
        bool r8583997 = r8583995 <= r8583996;
        double r8583998 = r8583995 * r8583995;
        double r8583999 = 4.0;
        double r8584000 = r8583998 * r8583999;
        double r8584001 = 1.0;
        double r8584002 = r8584000 - r8584001;
        double r8584003 = r8584002 * r8583999;
        double r8584004 = r8583998 / r8584003;
        double r8584005 = 0.00390625;
        double r8584006 = r8584005 / r8583998;
        double r8584007 = 0.015625;
        double r8584008 = r8584006 + r8584007;
        double r8584009 = r8584008 / r8583998;
        double r8584010 = 0.0625;
        double r8584011 = r8584009 + r8584010;
        double r8584012 = r8583997 ? r8584004 : r8584011;
        return r8584012;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 219.36859705155612

    1. Initial program 45.1

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]

    if 219.36859705155612 < i

    1. Initial program 47.0

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified30.7

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
    3. Using strategy rm

    4. Applied associate-/l*30.7

      \[\leadsto \color{blue}{\frac{i}{\frac{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}{i}}}\]

    5. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    6. Simplified0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 219.36859705155612:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(i \cdot i\right) \cdot 4 - 1.0\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.00390625}{i \cdot i} + 0.015625}{i \cdot i} + \frac{1}{16}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))