Average Error: 31.9 → 0.2
Time: 13.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\
\;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\end{array}
double f(double x) {
        double r897396 = 1.0;
        double r897397 = x;
        double r897398 = cos(r897397);
        double r897399 = r897396 - r897398;
        double r897400 = r897397 * r897397;
        double r897401 = r897399 / r897400;
        return r897401;
}
↓
double f(double x) {
        double r897402 = x;
        double r897403 = -0.03115379972147734;
        bool r897404 = r897402 <= r897403;
        double r897405 = 1.0;
        double r897406 = cos(r897402);
        double r897407 = r897405 - r897406;
        double r897408 = r897407 / r897402;
        double r897409 = r897408 / r897402;
        double r897410 = 0.027390056951785625;
        bool r897411 = r897402 <= r897410;
        double r897412 = 0.5;
        double r897413 = r897402 * r897402;
        double r897414 = 0.001388888888888889;
        double r897415 = r897413 * r897414;
        double r897416 = -0.041666666666666664;
        double r897417 = r897415 + r897416;
        double r897418 = r897413 * r897417;
        double r897419 = r897412 + r897418;
        double r897420 = r897411 ? r897419 : r897409;
        double r897421 = r897404 ? r897409 : r897420;
        return r897421;
}

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 < -0.03115379972147734 or 0.027390056951785625 < x

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied associate-*r/0.5

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

    7. Simplified0.5

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

    if -0.03115379972147734 < x < 0.027390056951785625

    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. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))