Average Error: 31.4 → 0.2
Time: 7.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029841445064018916 \lor \neg \left(x \le 0.0302568236777068268\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029841445064018916 \lor \neg \left(x \le 0.0302568236777068268\right):\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{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 r31393 = 1.0;
        double r31394 = x;
        double r31395 = cos(r31394);
        double r31396 = r31393 - r31395;
        double r31397 = r31394 * r31394;
        double r31398 = r31396 / r31397;
        return r31398;
}
↓
double f(double x) {
        double r31399 = x;
        double r31400 = -0.029841445064018916;
        bool r31401 = r31399 <= r31400;
        double r31402 = 0.030256823677706827;
        bool r31403 = r31399 <= r31402;
        double r31404 = !r31403;
        bool r31405 = r31401 || r31404;
        double r31406 = 1.0;
        double r31407 = cos(r31399);
        double r31408 = r31406 - r31407;
        double r31409 = r31408 / r31399;
        double r31410 = r31409 / r31399;
        double r31411 = 0.001388888888888889;
        double r31412 = 4.0;
        double r31413 = pow(r31399, r31412);
        double r31414 = r31411 * r31413;
        double r31415 = 0.5;
        double r31416 = r31414 + r31415;
        double r31417 = 0.041666666666666664;
        double r31418 = 2.0;
        double r31419 = pow(r31399, r31418);
        double r31420 = r31417 * r31419;
        double r31421 = r31416 - r31420;
        double r31422 = r31405 ? r31410 : r31421;
        return r31422;
}

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 < -0.029841445064018916 or 0.030256823677706827 < x

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied times-frac0.6

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

    6. Applied associate-*r/0.6

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

    7. Simplified0.5

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

    if -0.029841445064018916 < x < 0.030256823677706827

    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. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029841445064018916 \lor \neg \left(x \le 0.0302568236777068268\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))