Average Error: 31.4 → 0.2
Time: 4.1s
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 r23857 = 1.0;
        double r23858 = x;
        double r23859 = cos(r23858);
        double r23860 = r23857 - r23859;
        double r23861 = r23858 * r23858;
        double r23862 = r23860 / r23861;
        return r23862;
}
↓
double f(double x) {
        double r23863 = x;
        double r23864 = -0.03408999554764363;
        bool r23865 = r23863 <= r23864;
        double r23866 = 1.0;
        double r23867 = 3.0;
        double r23868 = pow(r23866, r23867);
        double r23869 = cos(r23863);
        double r23870 = pow(r23869, r23867);
        double r23871 = r23868 - r23870;
        double r23872 = exp(r23871);
        double r23873 = log(r23872);
        double r23874 = r23869 + r23866;
        double r23875 = r23869 * r23874;
        double r23876 = r23866 * r23866;
        double r23877 = r23875 + r23876;
        double r23878 = r23873 / r23877;
        double r23879 = r23878 / r23863;
        double r23880 = r23879 / r23863;
        double r23881 = 0.031210463015068444;
        bool r23882 = r23863 <= r23881;
        double r23883 = 0.001388888888888889;
        double r23884 = 4.0;
        double r23885 = pow(r23863, r23884);
        double r23886 = r23883 * r23885;
        double r23887 = 0.5;
        double r23888 = r23886 + r23887;
        double r23889 = 0.041666666666666664;
        double r23890 = 2.0;
        double r23891 = pow(r23863, r23890);
        double r23892 = r23889 * r23891;
        double r23893 = r23888 - r23892;
        double r23894 = r23866 - r23869;
        double r23895 = r23877 * r23894;
        double r23896 = r23895 / r23877;
        double r23897 = r23896 / r23863;
        double r23898 = r23897 / r23863;
        double r23899 = r23882 ? r23893 : r23898;
        double r23900 = r23865 ? r23880 : r23899;
        return r23900;
}

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)))