Average Error: 31.2 → 0.2
Time: 9.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\
\;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\end{array}
double f(double x) {
        double r29678 = 1.0;
        double r29679 = x;
        double r29680 = cos(r29679);
        double r29681 = r29678 - r29680;
        double r29682 = r29679 * r29679;
        double r29683 = r29681 / r29682;
        return r29683;
}
↓
double f(double x) {
        double r29684 = x;
        double r29685 = -0.03100648570835607;
        bool r29686 = r29684 <= r29685;
        double r29687 = 0.0325473048632644;
        bool r29688 = r29684 <= r29687;
        double r29689 = !r29688;
        bool r29690 = r29686 || r29689;
        double r29691 = 1.0;
        double r29692 = cos(r29684);
        double r29693 = r29691 - r29692;
        double r29694 = 1.0;
        double r29695 = r29694 / r29684;
        double r29696 = r29693 * r29695;
        double r29697 = r29696 / r29684;
        double r29698 = 0.001388888888888889;
        double r29699 = 4.0;
        double r29700 = pow(r29684, r29699);
        double r29701 = r29698 * r29700;
        double r29702 = 0.5;
        double r29703 = 0.041666666666666664;
        double r29704 = 2.0;
        double r29705 = pow(r29684, r29704);
        double r29706 = r29703 * r29705;
        double r29707 = r29702 - r29706;
        double r29708 = r29701 + r29707;
        double r29709 = r29690 ? r29697 : r29708;
        return r29709;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03100648570835607 or 0.0325473048632644 < x

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
    4. Using strategy rm

    5. Applied div-inv0.5

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

    if -0.03100648570835607 < x < 0.0325473048632644

    1. Initial program 62.2

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
    3. Using strategy rm

    4. Applied associate--l+0.0

      \[\leadsto \color{blue}{\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03100648570835606854245725116925314068794 \lor \neg \left(x \le 0.03254730486326439659050535624373878818005\right):\\ \;\;\;\;\frac{\left(1 - \cos x\right) \cdot \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))