Average Error: 31.4 → 0.2
Time: 4.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031682751964378281 \lor \neg \left(x \le 0.033639018339020957\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1 - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031682751964378281 \lor \neg \left(x \le 0.033639018339020957\right):\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{1 - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{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 r20253 = 1.0;
        double r20254 = x;
        double r20255 = cos(r20254);
        double r20256 = r20253 - r20255;
        double r20257 = r20254 * r20254;
        double r20258 = r20256 / r20257;
        return r20258;
}
↓
double f(double x) {
        double r20259 = x;
        double r20260 = -0.03168275196437828;
        bool r20261 = r20259 <= r20260;
        double r20262 = 0.03363901833902096;
        bool r20263 = r20259 <= r20262;
        double r20264 = !r20263;
        bool r20265 = r20261 || r20264;
        double r20266 = 1.0;
        double r20267 = r20266 / r20259;
        double r20268 = 1.0;
        double r20269 = cos(r20259);
        double r20270 = 3.0;
        double r20271 = pow(r20269, r20270);
        double r20272 = r20268 - r20271;
        double r20273 = r20269 + r20268;
        double r20274 = r20269 * r20273;
        double r20275 = r20268 * r20268;
        double r20276 = r20274 + r20275;
        double r20277 = r20272 / r20276;
        double r20278 = r20277 / r20259;
        double r20279 = r20267 * r20278;
        double r20280 = 0.001388888888888889;
        double r20281 = 4.0;
        double r20282 = pow(r20259, r20281);
        double r20283 = r20280 * r20282;
        double r20284 = 0.5;
        double r20285 = r20283 + r20284;
        double r20286 = 0.041666666666666664;
        double r20287 = 2.0;
        double r20288 = pow(r20259, r20287);
        double r20289 = r20286 * r20288;
        double r20290 = r20285 - r20289;
        double r20291 = r20265 ? r20279 : r20290;
        return r20291;
}

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 < -0.03168275196437828 or 0.03363901833902096 < 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-frac0.4

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

    6. Applied flip3--0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x}\]

    7. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x}\]
    8. Using strategy rm

    9. Applied cube-mult0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{1}^{3} - \color{blue}{\cos x \cdot \left(\cos x \cdot \cos x\right)}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\]

    10. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{{1}^{3} - \cos x \cdot \color{blue}{{\left(\cos x\right)}^{2}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\]

    11. Taylor expanded around inf 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{\color{blue}{1 - {\left(\cos x\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\]

    if -0.03168275196437828 < x < 0.03363901833902096

    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. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031682751964378281 \lor \neg \left(x \le 0.033639018339020957\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1 - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))