Average Error: 31.7 → 0.3
Time: 4.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03396454979348009078909953473157656844705:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03396454979348009078909953473157656844705:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\

\mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\

\end{array}
double f(double x) {
        double r31199 = 1.0;
        double r31200 = x;
        double r31201 = cos(r31200);
        double r31202 = r31199 - r31201;
        double r31203 = r31200 * r31200;
        double r31204 = r31202 / r31203;
        return r31204;
}
↓
double f(double x) {
        double r31205 = x;
        double r31206 = -0.03396454979348009;
        bool r31207 = r31205 <= r31206;
        double r31208 = 1.0;
        double r31209 = r31208 / r31205;
        double r31210 = 1.0;
        double r31211 = 3.0;
        double r31212 = pow(r31210, r31211);
        double r31213 = cos(r31205);
        double r31214 = pow(r31213, r31211);
        double r31215 = exp(r31214);
        double r31216 = log(r31215);
        double r31217 = r31212 - r31216;
        double r31218 = 2.0;
        double r31219 = pow(r31213, r31218);
        double r31220 = r31210 * r31210;
        double r31221 = r31219 - r31220;
        double r31222 = r31213 - r31210;
        double r31223 = r31221 / r31222;
        double r31224 = r31213 * r31223;
        double r31225 = r31224 + r31220;
        double r31226 = r31217 / r31225;
        double r31227 = r31226 / r31205;
        double r31228 = r31209 * r31227;
        double r31229 = 0.031416007424221945;
        bool r31230 = r31205 <= r31229;
        double r31231 = 0.001388888888888889;
        double r31232 = 4.0;
        double r31233 = pow(r31205, r31232);
        double r31234 = r31231 * r31233;
        double r31235 = 0.5;
        double r31236 = r31234 + r31235;
        double r31237 = 0.041666666666666664;
        double r31238 = pow(r31205, r31218);
        double r31239 = r31237 * r31238;
        double r31240 = r31236 - r31239;
        double r31241 = pow(r31214, r31211);
        double r31242 = cbrt(r31241);
        double r31243 = r31212 - r31242;
        double r31244 = r31213 + r31210;
        double r31245 = r31213 * r31244;
        double r31246 = r31245 + r31220;
        double r31247 = r31243 / r31246;
        double r31248 = r31247 / r31205;
        double r31249 = r31209 * r31248;
        double r31250 = r31230 ? r31240 : r31249;
        double r31251 = r31207 ? r31228 : r31250;
        return r31251;
}

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 < -0.03396454979348009

    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-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
    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 add-log-exp0.5

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

    11. Applied flip-+0.5

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

    12. Simplified0.5

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

    if -0.03396454979348009 < x < 0.031416007424221945

    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 0.031416007424221945 < x

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity1.2

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

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
    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 add-log-exp0.5

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

    11. Applied add-cbrt-cube0.6

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

    12. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03396454979348009078909953473157656844705:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{x}\\ \mathbf{elif}\;x \le 0.03141600742422194503244980978706735186279:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{1}^{3} - \sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))