Average Error: 30.8 → 0.5
Time: 4.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03437938882255739403426275657693622633815:\\ \;\;\;\;\frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0311908640075921408940651247121422784403:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03437938882255739403426275657693622633815:\\
\;\;\;\;\frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}}{x}\\

\mathbf{elif}\;x \le 0.0311908640075921408940651247121422784403:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r24317 = 1.0;
        double r24318 = x;
        double r24319 = cos(r24318);
        double r24320 = r24317 - r24319;
        double r24321 = r24318 * r24318;
        double r24322 = r24320 / r24321;
        return r24322;
}
↓
double f(double x) {
        double r24323 = x;
        double r24324 = -0.034379388822557394;
        bool r24325 = r24323 <= r24324;
        double r24326 = 1.0;
        double r24327 = r24326 * r24326;
        double r24328 = cos(r24323);
        double r24329 = r24328 * r24328;
        double r24330 = r24327 - r24329;
        double r24331 = r24326 + r24328;
        double r24332 = r24323 * r24331;
        double r24333 = r24330 / r24332;
        double r24334 = r24333 / r24323;
        double r24335 = 0.03119086400759214;
        bool r24336 = r24323 <= r24335;
        double r24337 = 0.001388888888888889;
        double r24338 = 4.0;
        double r24339 = pow(r24323, r24338);
        double r24340 = r24337 * r24339;
        double r24341 = 0.5;
        double r24342 = r24340 + r24341;
        double r24343 = 0.041666666666666664;
        double r24344 = 2.0;
        double r24345 = pow(r24323, r24344);
        double r24346 = r24343 * r24345;
        double r24347 = r24342 - r24346;
        double r24348 = 3.0;
        double r24349 = pow(r24326, r24348);
        double r24350 = pow(r24328, r24348);
        double r24351 = r24349 - r24350;
        double r24352 = r24328 + r24326;
        double r24353 = r24328 * r24352;
        double r24354 = r24353 + r24327;
        double r24355 = r24351 / r24354;
        double r24356 = r24323 * r24323;
        double r24357 = r24355 / r24356;
        double r24358 = r24336 ? r24347 : r24357;
        double r24359 = r24325 ? r24334 : r24358;
        return r24359;
}

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 < -0.034379388822557394

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied times-frac0.5

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

    6. Applied associate-*r/0.5

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

    7. Simplified0.4

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

    9. Applied flip--0.7

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

    10. Applied associate-/l/0.7

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

    if -0.034379388822557394 < x < 0.03119086400759214

    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 0.03119086400759214 < x

    1. Initial program 1.1

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

    3. Applied flip3--1.1

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

    4. Simplified1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03437938882255739403426275657693622633815:\\ \;\;\;\;\frac{\frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.0311908640075921408940651247121422784403:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))