Average Error: 31.2 → 0.4
Time: 16.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\ \mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\

\mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r20210 = 1.0;
        double r20211 = x;
        double r20212 = cos(r20211);
        double r20213 = r20210 - r20212;
        double r20214 = r20211 * r20211;
        double r20215 = r20213 / r20214;
        return r20215;
}
↓
double f(double x) {
        double r20216 = x;
        double r20217 = -0.030846791202208904;
        bool r20218 = r20216 <= r20217;
        double r20219 = 1.0;
        double r20220 = cos(r20216);
        double r20221 = r20219 - r20220;
        double r20222 = sqrt(r20221);
        double r20223 = r20222 / r20216;
        double r20224 = r20216 / r20223;
        double r20225 = r20222 / r20224;
        double r20226 = 0.03981852127604897;
        bool r20227 = r20216 <= r20226;
        double r20228 = 0.001388888888888889;
        double r20229 = 4.0;
        double r20230 = pow(r20216, r20229);
        double r20231 = r20228 * r20230;
        double r20232 = 0.5;
        double r20233 = r20231 + r20232;
        double r20234 = 0.041666666666666664;
        double r20235 = 2.0;
        double r20236 = pow(r20216, r20235);
        double r20237 = r20234 * r20236;
        double r20238 = r20233 - r20237;
        double r20239 = r20219 / r20216;
        double r20240 = r20220 / r20216;
        double r20241 = r20239 - r20240;
        double r20242 = r20241 / r20216;
        double r20243 = r20227 ? r20238 : r20242;
        double r20244 = r20218 ? r20225 : r20243;
        return r20244;
}

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 < -0.030846791202208904

    1. Initial program 0.9

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

    3. Applied associate-/r*0.5

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

    5. Applied *-un-lft-identity0.5

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

    6. Applied add-sqr-sqrt0.6

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

    7. Applied times-frac0.6

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

    8. Applied associate-/l*1.1

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

    if -0.030846791202208904 < x < 0.03981852127604897

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.03981852127604897 < x

    1. Initial program 1.1

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

    3. Applied associate-/r*0.4

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

    5. Applied div-sub0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\ \mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))