Average Error: 31.8 → 0.2
Time: 15.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03306643978524941018815042070855270139873 \lor \neg \left(x \le 0.02855355443278343591240187038238218519837\right):\\ \;\;\;\;\frac{-\frac{\cos x - 1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03306643978524941018815042070855270139873 \lor \neg \left(x \le 0.02855355443278343591240187038238218519837\right):\\
\;\;\;\;\frac{-\frac{\cos x - 1}{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 r18075 = 1.0;
        double r18076 = x;
        double r18077 = cos(r18076);
        double r18078 = r18075 - r18077;
        double r18079 = r18076 * r18076;
        double r18080 = r18078 / r18079;
        return r18080;
}
↓
double f(double x) {
        double r18081 = x;
        double r18082 = -0.03306643978524941;
        bool r18083 = r18081 <= r18082;
        double r18084 = 0.028553554432783436;
        bool r18085 = r18081 <= r18084;
        double r18086 = !r18085;
        bool r18087 = r18083 || r18086;
        double r18088 = cos(r18081);
        double r18089 = 1.0;
        double r18090 = r18088 - r18089;
        double r18091 = r18090 / r18081;
        double r18092 = -r18091;
        double r18093 = r18092 / r18081;
        double r18094 = 0.001388888888888889;
        double r18095 = 4.0;
        double r18096 = pow(r18081, r18095);
        double r18097 = r18094 * r18096;
        double r18098 = 0.5;
        double r18099 = r18097 + r18098;
        double r18100 = 0.041666666666666664;
        double r18101 = 2.0;
        double r18102 = pow(r18081, r18101);
        double r18103 = r18100 * r18102;
        double r18104 = r18099 - r18103;
        double r18105 = r18087 ? r18093 : r18104;
        return r18105;
}

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 < -0.03306643978524941 or 0.028553554432783436 < x

    1. Initial program 1.0

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

    3. Applied associate-/r*0.5

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

    4. Taylor expanded around -inf 0.5

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

    5. Simplified0.5

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

    if -0.03306643978524941 < x < 0.028553554432783436

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03306643978524941018815042070855270139873 \lor \neg \left(x \le 0.02855355443278343591240187038238218519837\right):\\ \;\;\;\;\frac{-\frac{\cos x - 1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))