Average Error: 31.1 → 0.2
Time: 10.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029542526183904051217954389585429453291 \lor \neg \left(x \le 0.03047467434030125613131367856567521812394\right):\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029542526183904051217954389585429453291 \lor \neg \left(x \le 0.03047467434030125613131367856567521812394\right):\\
\;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\

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

\end{array}
double f(double x) {
        double r17758 = 1.0;
        double r17759 = x;
        double r17760 = cos(r17759);
        double r17761 = r17758 - r17760;
        double r17762 = r17759 * r17759;
        double r17763 = r17761 / r17762;
        return r17763;
}
↓
double f(double x) {
        double r17764 = x;
        double r17765 = -0.02954252618390405;
        bool r17766 = r17764 <= r17765;
        double r17767 = 0.030474674340301256;
        bool r17768 = r17764 <= r17767;
        double r17769 = !r17768;
        bool r17770 = r17766 || r17769;
        double r17771 = 1.0;
        double r17772 = r17771 / r17764;
        double r17773 = 1.0;
        double r17774 = cos(r17764);
        double r17775 = r17773 - r17774;
        double r17776 = r17772 * r17775;
        double r17777 = r17776 / r17764;
        double r17778 = 0.001388888888888889;
        double r17779 = 4.0;
        double r17780 = pow(r17764, r17779);
        double r17781 = r17778 * r17780;
        double r17782 = 0.5;
        double r17783 = r17781 + r17782;
        double r17784 = 0.041666666666666664;
        double r17785 = 2.0;
        double r17786 = pow(r17764, r17785);
        double r17787 = r17784 * r17786;
        double r17788 = r17783 - r17787;
        double r17789 = r17770 ? r17777 : r17788;
        return r17789;
}

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 < -0.02954252618390405 or 0.030474674340301256 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip--0.7

      \[\leadsto \frac{\frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x}}{x}\]

    6. Applied associate-/l/0.7

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}}}{x}\]
    7. Using strategy rm

    8. Applied flip-+0.5

      \[\leadsto \frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 - \cos x}}}}{x}\]

    9. Applied associate-*r/0.5

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

    10. Applied associate-/r/0.5

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

    11. Simplified0.5

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

    if -0.02954252618390405 < x < 0.030474674340301256

    1. Initial program 62.4

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029542526183904051217954389585429453291 \lor \neg \left(x \le 0.03047467434030125613131367856567521812394\right):\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))