Average Error: 31.3 → 0.4
Time: 13.5s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03428006622296721206399894299465813674033:\\ \;\;\;\;\frac{\frac{\frac{{1}^{4} - {\left(\cos x\right)}^{4}}{{\left(\cos x\right)}^{2} + 1 \cdot 1}}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.02235098969897789805694188203233352396637:\\ \;\;\;\;\left(\frac{-1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03428006622296721206399894299465813674033:\\
\;\;\;\;\frac{\frac{\frac{{1}^{4} - {\left(\cos x\right)}^{4}}{{\left(\cos x\right)}^{2} + 1 \cdot 1}}{x \cdot \left(1 + \cos x\right)}}{x}\\

\mathbf{elif}\;x \le 0.02235098969897789805694188203233352396637:\\
\;\;\;\;\left(\frac{-1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r19308 = 1.0;
        double r19309 = x;
        double r19310 = cos(r19309);
        double r19311 = r19308 - r19310;
        double r19312 = r19309 * r19309;
        double r19313 = r19311 / r19312;
        return r19313;
}
↓
double f(double x) {
        double r19314 = x;
        double r19315 = -0.03428006622296721;
        bool r19316 = r19314 <= r19315;
        double r19317 = 1.0;
        double r19318 = 4.0;
        double r19319 = pow(r19317, r19318);
        double r19320 = cos(r19314);
        double r19321 = pow(r19320, r19318);
        double r19322 = r19319 - r19321;
        double r19323 = 2.0;
        double r19324 = pow(r19320, r19323);
        double r19325 = r19317 * r19317;
        double r19326 = r19324 + r19325;
        double r19327 = r19322 / r19326;
        double r19328 = r19317 + r19320;
        double r19329 = r19314 * r19328;
        double r19330 = r19327 / r19329;
        double r19331 = r19330 / r19314;
        double r19332 = 0.022350989698977898;
        bool r19333 = r19314 <= r19332;
        double r19334 = -0.041666666666666664;
        double r19335 = r19314 * r19314;
        double r19336 = 0.001388888888888889;
        double r19337 = r19335 * r19336;
        double r19338 = r19334 + r19337;
        double r19339 = r19338 * r19335;
        double r19340 = 0.5;
        double r19341 = r19339 + r19340;
        double r19342 = r19317 - r19320;
        double r19343 = r19342 / r19335;
        double r19344 = r19333 ? r19341 : r19343;
        double r19345 = r19316 ? r19331 : r19344;
        return r19345;
}

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 < -0.03428006622296721

    1. Initial program 1.2

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip--0.7

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

    6. Applied associate-/l/0.7

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

    8. Applied flip--0.8

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

    9. Simplified0.7

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

    10. Simplified0.7

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

    if -0.03428006622296721 < x < 0.022350989698977898

    1. Initial program 62.3

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

    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. Simplified0.0

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

    if 0.022350989698977898 < x

    1. Initial program 1.0

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03428006622296721206399894299465813674033:\\ \;\;\;\;\frac{\frac{\frac{{1}^{4} - {\left(\cos x\right)}^{4}}{{\left(\cos x\right)}^{2} + 1 \cdot 1}}{x \cdot \left(1 + \cos x\right)}}{x}\\ \mathbf{elif}\;x \le 0.02235098969897789805694188203233352396637:\\ \;\;\;\;\left(\frac{-1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(x \cdot x\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))