Average Error: 31.3 → 0.3
Time: 4.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{elif}\;x \le 0.0246736691289138228:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031288658239448007:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\

\mathbf{elif}\;x \le 0.0246736691289138228:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r25414 = 1.0;
        double r25415 = x;
        double r25416 = cos(r25415);
        double r25417 = r25414 - r25416;
        double r25418 = r25415 * r25415;
        double r25419 = r25417 / r25418;
        return r25419;
}
↓
double f(double x) {
        double r25420 = x;
        double r25421 = -0.03128865823944801;
        bool r25422 = r25420 <= r25421;
        double r25423 = 1.0;
        double r25424 = r25423 / r25420;
        double r25425 = 1.0;
        double r25426 = cos(r25420);
        double r25427 = r25425 - r25426;
        double r25428 = r25427 / r25420;
        double r25429 = r25424 * r25428;
        double r25430 = 0.024673669128913823;
        bool r25431 = r25420 <= r25430;
        double r25432 = 0.001388888888888889;
        double r25433 = 4.0;
        double r25434 = pow(r25420, r25433);
        double r25435 = r25432 * r25434;
        double r25436 = 0.5;
        double r25437 = r25435 + r25436;
        double r25438 = 0.041666666666666664;
        double r25439 = 2.0;
        double r25440 = pow(r25420, r25439);
        double r25441 = r25438 * r25440;
        double r25442 = r25437 - r25441;
        double r25443 = r25420 / r25427;
        double r25444 = r25423 / r25443;
        double r25445 = r25424 * r25444;
        double r25446 = r25431 ? r25442 : r25445;
        double r25447 = r25422 ? r25429 : r25446;
        return r25447;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03128865823944801

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied times-frac0.5

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

    if -0.03128865823944801 < x < 0.024673669128913823

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

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

    if 0.024673669128913823 < x

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.6

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

    6. Applied clear-num0.6

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{elif}\;x \le 0.0246736691289138228:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))