Average Error: 31.5 → 0.5
Time: 10.2s
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 r30890 = 1.0;
        double r30891 = x;
        double r30892 = cos(r30891);
        double r30893 = r30890 - r30892;
        double r30894 = r30891 * r30891;
        double r30895 = r30893 / r30894;
        return r30895;
}
↓
double f(double x) {
        double r30896 = x;
        double r30897 = -0.02915897731694385;
        bool r30898 = r30896 <= r30897;
        double r30899 = 1.0;
        double r30900 = r30899 / r30896;
        double r30901 = 1.0;
        double r30902 = 3.0;
        double r30903 = pow(r30901, r30902);
        double r30904 = cos(r30896);
        double r30905 = pow(r30904, r30902);
        double r30906 = r30903 - r30905;
        double r30907 = r30896 / r30906;
        double r30908 = r30901 * r30901;
        double r30909 = 0.5;
        double r30910 = 2.0;
        double r30911 = r30910 * r30896;
        double r30912 = cos(r30911);
        double r30913 = r30909 * r30912;
        double r30914 = r30901 * r30904;
        double r30915 = r30913 + r30914;
        double r30916 = r30909 + r30915;
        double r30917 = r30908 + r30916;
        double r30918 = r30907 * r30917;
        double r30919 = r30900 / r30918;
        double r30920 = 0.03476390489487963;
        bool r30921 = r30896 <= r30920;
        double r30922 = 0.00138888888888887;
        double r30923 = 4.0;
        double r30924 = pow(r30896, r30923);
        double r30925 = r30922 * r30924;
        double r30926 = 0.5;
        double r30927 = r30925 + r30926;
        double r30928 = 0.041666666666666685;
        double r30929 = pow(r30896, r30910);
        double r30930 = r30928 * r30929;
        double r30931 = r30927 - r30930;
        double r30932 = r30896 * r30896;
        double r30933 = r30901 / r30932;
        double r30934 = r30904 / r30932;
        double r30935 = r30933 - r30934;
        double r30936 = r30921 ? r30931 : r30935;
        double r30937 = r30898 ? r30919 : r30936;
        return r30937;
}

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)))