Average Error: 32.0 → 0.2
Time: 13.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{720} \cdot {x}^{4}\right) + \frac{1}{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

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

\end{array}
double f(double x) {
        double r21932 = 1.0;
        double r21933 = x;
        double r21934 = cos(r21933);
        double r21935 = r21932 - r21934;
        double r21936 = r21933 * r21933;
        double r21937 = r21935 / r21936;
        return r21937;
}
↓
double f(double x) {
        double r21938 = x;
        double r21939 = -0.029468940558385545;
        bool r21940 = r21938 <= r21939;
        double r21941 = 0.029870589015516296;
        bool r21942 = r21938 <= r21941;
        double r21943 = !r21942;
        bool r21944 = r21940 || r21943;
        double r21945 = 1.0;
        double r21946 = cos(r21938);
        double r21947 = r21945 - r21946;
        double r21948 = r21947 / r21938;
        double r21949 = r21948 / r21938;
        double r21950 = -0.041666666666666664;
        double r21951 = 2.0;
        double r21952 = pow(r21938, r21951);
        double r21953 = r21950 * r21952;
        double r21954 = 0.001388888888888889;
        double r21955 = 4.0;
        double r21956 = pow(r21938, r21955);
        double r21957 = r21954 * r21956;
        double r21958 = r21953 + r21957;
        double r21959 = 0.5;
        double r21960 = r21958 + r21959;
        double r21961 = r21944 ? r21949 : r21960;
        return r21961;
}

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 < -0.029468940558385545 or 0.029870589015516296 < x

    1. Initial program 1.0

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

    3. Applied associate-/r*0.5

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

    if -0.029468940558385545 < x < 0.029870589015516296

    1. Initial program 62.2

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

    3. Applied associate-/r*61.3

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{24} \cdot {x}^{2} + \frac{1}{720} \cdot {x}^{4}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))