Average Error: 31.4 → 0.2
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03408999554764362976966296514547138940543:\\ \;\;\;\;\frac{\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03121046301506844380946326111825328553095:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03408999554764362976966296514547138940543:\\
\;\;\;\;\frac{\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\

\mathbf{elif}\;x \le 0.03121046301506844380946326111825328553095:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\

\end{array}
double f(double x) {
        double r24244 = 1.0;
        double r24245 = x;
        double r24246 = cos(r24245);
        double r24247 = r24244 - r24246;
        double r24248 = r24245 * r24245;
        double r24249 = r24247 / r24248;
        return r24249;
}
↓
double f(double x) {
        double r24250 = x;
        double r24251 = -0.03408999554764363;
        bool r24252 = r24250 <= r24251;
        double r24253 = 1.0;
        double r24254 = 3.0;
        double r24255 = pow(r24253, r24254);
        double r24256 = cos(r24250);
        double r24257 = pow(r24256, r24254);
        double r24258 = r24255 - r24257;
        double r24259 = exp(r24258);
        double r24260 = log(r24259);
        double r24261 = r24256 + r24253;
        double r24262 = r24256 * r24261;
        double r24263 = r24253 * r24253;
        double r24264 = r24262 + r24263;
        double r24265 = r24260 / r24264;
        double r24266 = r24265 / r24250;
        double r24267 = r24266 / r24250;
        double r24268 = 0.031210463015068444;
        bool r24269 = r24250 <= r24268;
        double r24270 = 0.001388888888888889;
        double r24271 = 4.0;
        double r24272 = pow(r24250, r24271);
        double r24273 = r24270 * r24272;
        double r24274 = 0.5;
        double r24275 = r24273 + r24274;
        double r24276 = 0.041666666666666664;
        double r24277 = 2.0;
        double r24278 = pow(r24250, r24277);
        double r24279 = r24276 * r24278;
        double r24280 = r24275 - r24279;
        double r24281 = r24253 - r24256;
        double r24282 = r24264 * r24281;
        double r24283 = r24282 / r24264;
        double r24284 = r24283 / r24250;
        double r24285 = r24284 / r24250;
        double r24286 = r24269 ? r24280 : r24285;
        double r24287 = r24252 ? r24267 : r24286;
        return r24287;
}

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 < -0.03408999554764363

    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 flip3--0.5

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

    6. Simplified0.5

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

    8. Applied add-log-exp0.5

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

    9. Applied add-log-exp0.5

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

    10. Applied diff-log0.6

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

    11. Simplified0.5

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

    if -0.03408999554764363 < x < 0.031210463015068444

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.0

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

    if 0.031210463015068444 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip3--0.5

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

    6. Simplified0.5

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

    8. Applied difference-cubes0.5

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

    9. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03408999554764362976966296514547138940543:\\ \;\;\;\;\frac{\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \mathbf{elif}\;x \le 0.03121046301506844380946326111825328553095:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))