Average Error: 31.5 → 0.5
Time: 9.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029158977316943849:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)\right)}\\ \mathbf{elif}\;x \le 0.034763904894879627:\\ \;\;\;\;\left(0.00138888888888887 \cdot {x}^{4} + 0.5\right) - 0.041666666666666685 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029158977316943849:\\
\;\;\;\;\frac{\frac{1}{x}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)\right)}\\

\mathbf{elif}\;x \le 0.034763904894879627:\\
\;\;\;\;\left(0.00138888888888887 \cdot {x}^{4} + 0.5\right) - 0.041666666666666685 \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r30888 = 1.0;
        double r30889 = x;
        double r30890 = cos(r30889);
        double r30891 = r30888 - r30890;
        double r30892 = r30889 * r30889;
        double r30893 = r30891 / r30892;
        return r30893;
}
↓
double f(double x) {
        double r30894 = x;
        double r30895 = -0.02915897731694385;
        bool r30896 = r30894 <= r30895;
        double r30897 = 1.0;
        double r30898 = r30897 / r30894;
        double r30899 = 1.0;
        double r30900 = 3.0;
        double r30901 = pow(r30899, r30900);
        double r30902 = cos(r30894);
        double r30903 = pow(r30902, r30900);
        double r30904 = r30901 - r30903;
        double r30905 = r30894 / r30904;
        double r30906 = r30899 * r30899;
        double r30907 = 0.5;
        double r30908 = 2.0;
        double r30909 = r30908 * r30894;
        double r30910 = cos(r30909);
        double r30911 = r30907 * r30910;
        double r30912 = r30899 * r30902;
        double r30913 = r30911 + r30912;
        double r30914 = r30907 + r30913;
        double r30915 = r30906 + r30914;
        double r30916 = r30905 * r30915;
        double r30917 = r30898 / r30916;
        double r30918 = 0.03476390489487963;
        bool r30919 = r30894 <= r30918;
        double r30920 = 0.00138888888888887;
        double r30921 = 4.0;
        double r30922 = pow(r30894, r30921);
        double r30923 = r30920 * r30922;
        double r30924 = 0.5;
        double r30925 = r30923 + r30924;
        double r30926 = 0.041666666666666685;
        double r30927 = pow(r30894, r30908);
        double r30928 = r30926 * r30927;
        double r30929 = r30925 - r30928;
        double r30930 = r30894 * r30894;
        double r30931 = r30899 / r30930;
        double r30932 = r30902 / r30930;
        double r30933 = r30931 - r30932;
        double r30934 = r30919 ? r30929 : r30933;
        double r30935 = r30896 ? r30917 : r30934;
        return r30935;
}

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 < -0.02915897731694385

    1. Initial program 1.1

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

    3. Applied clear-num1.2

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

    4. Simplified1.1

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

    6. Applied flip3--1.2

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

    7. Applied associate-/r/1.2

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

    8. Applied associate-*r*1.2

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

    10. Applied sqr-cos1.2

      \[\leadsto \frac{1}{\left(x \cdot \frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)} + 1 \cdot \cos x\right)\right)}\]

    11. Applied associate-+l+1.2

      \[\leadsto \frac{1}{\left(x \cdot \frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \color{blue}{\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)}\right)}\]
    12. Using strategy rm

    13. Applied div-inv1.2

      \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(x \cdot \frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)\right)}}\]

    14. Simplified0.6

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{x}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)\right)}}\]

    if -0.02915897731694385 < x < 0.03476390489487963

    1. Initial program 62.2

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

    3. Applied clear-num62.2

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

    4. Simplified61.3

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

    6. Applied flip3--61.3

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

    7. Applied associate-/r/61.3

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

    8. Applied associate-*r*61.3

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

    10. Applied sqr-cos61.3

      \[\leadsto \frac{1}{\left(x \cdot \frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)} + 1 \cdot \cos x\right)\right)}\]

    11. Applied associate-+l+61.3

      \[\leadsto \frac{1}{\left(x \cdot \frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \color{blue}{\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)}\right)}\]

    12. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(0.00138888888888887 \cdot {x}^{4} + 0.5\right) - 0.041666666666666685 \cdot {x}^{2}}\]

    if 0.03476390489487963 < x

    1. Initial program 1.2

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

    3. Applied div-sub1.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029158977316943849:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right) + 1 \cdot \cos x\right)\right)\right)}\\ \mathbf{elif}\;x \le 0.034763904894879627:\\ \;\;\;\;\left(0.00138888888888887 \cdot {x}^{4} + 0.5\right) - 0.041666666666666685 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))