Average Error: 31.5 → 0.5
Time: 12.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-142128397306321}{4503599627370496} \lor \neg \left(x \le \frac{492456964496497}{18014398509481984}\right):\\ \;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{x}^{4}}{720} + \frac{1}{2}\right) - \frac{{x}^{2}}{24}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-142128397306321}{4503599627370496} \lor \neg \left(x \le \frac{492456964496497}{18014398509481984}\right):\\
\;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\

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

\end{array}
double f(double x) {
        double r17790 = 1.0;
        double r17791 = x;
        double r17792 = cos(r17791);
        double r17793 = r17790 - r17792;
        double r17794 = r17791 * r17791;
        double r17795 = r17793 / r17794;
        return r17795;
}
↓
double f(double x) {
        double r17796 = x;
        double r17797 = -142128397306321.0;
        double r17798 = 4503599627370496.0;
        double r17799 = r17797 / r17798;
        bool r17800 = r17796 <= r17799;
        double r17801 = 492456964496497.0;
        double r17802 = 18014398509481984.0;
        double r17803 = r17801 / r17802;
        bool r17804 = r17796 <= r17803;
        double r17805 = !r17804;
        bool r17806 = r17800 || r17805;
        double r17807 = 1.0;
        double r17808 = cos(r17796);
        double r17809 = r17807 - r17808;
        double r17810 = 2.0;
        double r17811 = pow(r17796, r17810);
        double r17812 = r17809 / r17811;
        double r17813 = 4.0;
        double r17814 = pow(r17796, r17813);
        double r17815 = 720.0;
        double r17816 = r17814 / r17815;
        double r17817 = 1.0;
        double r17818 = r17817 / r17810;
        double r17819 = r17816 + r17818;
        double r17820 = 24.0;
        double r17821 = r17811 / r17820;
        double r17822 = r17819 - r17821;
        double r17823 = r17806 ? r17812 : r17822;
        return r17823;
}

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 < -0.03155884382850993

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    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--0.6

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

    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 difference-of-squares0.5

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

    9. Applied associate-/l*0.5

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

    11. Applied add-exp-log0.5

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

    if -0.03155884382850993 < x < 0.02733685303105121

    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

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

    if 0.02733685303105121 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 *-un-lft-identity0.5

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

    6. Applied div-inv0.5

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

    7. Applied times-frac0.5

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

    8. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-142128397306321}{4503599627370496} \lor \neg \left(x \le \frac{492456964496497}{18014398509481984}\right):\\ \;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{x}^{4}}{720} + \frac{1}{2}\right) - \frac{{x}^{2}}{24}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))