Average Error: 31.9 → 0.4
Time: 3.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0327110828731407635 \lor \neg \left(x \le 0.031736096784144671\right):\\ \;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0327110828731407635 \lor \neg \left(x \le 0.031736096784144671\right):\\
\;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos 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 r18341 = 1.0;
        double r18342 = x;
        double r18343 = cos(r18342);
        double r18344 = r18341 - r18343;
        double r18345 = r18342 * r18342;
        double r18346 = r18344 / r18345;
        return r18346;
}
↓
double f(double x) {
        double r18347 = x;
        double r18348 = -0.032711082873140764;
        bool r18349 = r18347 <= r18348;
        double r18350 = 0.03173609678414467;
        bool r18351 = r18347 <= r18350;
        double r18352 = !r18351;
        bool r18353 = r18349 || r18352;
        double r18354 = 1.0;
        double r18355 = cos(r18347);
        double r18356 = r18354 - r18355;
        double r18357 = cbrt(r18356);
        double r18358 = r18357 * r18357;
        double r18359 = r18358 / r18347;
        double r18360 = r18357 / r18347;
        double r18361 = r18359 * r18360;
        double r18362 = 0.001388888888888889;
        double r18363 = 4.0;
        double r18364 = pow(r18347, r18363);
        double r18365 = r18362 * r18364;
        double r18366 = 0.5;
        double r18367 = r18365 + r18366;
        double r18368 = 0.041666666666666664;
        double r18369 = 2.0;
        double r18370 = pow(r18347, r18369);
        double r18371 = r18368 * r18370;
        double r18372 = r18367 - r18371;
        double r18373 = r18353 ? r18361 : r18372;
        return r18373;
}

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 < -0.032711082873140764 or 0.03173609678414467 < x

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt1.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}{x \cdot x}\]

    4. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}}\]

    if -0.032711082873140764 < x < 0.03173609678414467

    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0327110828731407635 \lor \neg \left(x \le 0.031736096784144671\right):\\ \;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))