Average Error: 31.4 → 0.3
Time: 4.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031454960231223988 \lor \neg \left(x \le 0.033122201796403697\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031454960231223988 \lor \neg \left(x \le 0.033122201796403697\right):\\
\;\;\;\;\frac{1}{x} \cdot \frac{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 r20429 = 1.0;
        double r20430 = x;
        double r20431 = cos(r20430);
        double r20432 = r20429 - r20431;
        double r20433 = r20430 * r20430;
        double r20434 = r20432 / r20433;
        return r20434;
}
↓
double f(double x) {
        double r20435 = x;
        double r20436 = -0.03145496023122399;
        bool r20437 = r20435 <= r20436;
        double r20438 = 0.0331222017964037;
        bool r20439 = r20435 <= r20438;
        double r20440 = !r20439;
        bool r20441 = r20437 || r20440;
        double r20442 = 1.0;
        double r20443 = r20442 / r20435;
        double r20444 = 1.0;
        double r20445 = cos(r20435);
        double r20446 = r20444 - r20445;
        double r20447 = r20446 / r20435;
        double r20448 = r20443 * r20447;
        double r20449 = 0.001388888888888889;
        double r20450 = 4.0;
        double r20451 = pow(r20435, r20450);
        double r20452 = r20449 * r20451;
        double r20453 = 0.5;
        double r20454 = r20452 + r20453;
        double r20455 = 0.041666666666666664;
        double r20456 = 2.0;
        double r20457 = pow(r20435, r20456);
        double r20458 = r20455 * r20457;
        double r20459 = r20454 - r20458;
        double r20460 = r20441 ? r20448 : r20459;
        return r20460;
}

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 < -0.03145496023122399 or 0.0331222017964037 < x

    1. Initial program 1.0

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

    3. Applied add-exp-log1.0

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x \cdot x}\]
    4. Using strategy rm

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

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

    6. Applied log-prod1.0

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

    7. Applied exp-sum1.0

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

    8. Applied times-frac0.5

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

    9. Simplified0.5

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

    10. Simplified0.5

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

    if -0.03145496023122399 < x < 0.0331222017964037

    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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031454960231223988 \lor \neg \left(x \le 0.033122201796403697\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))