Average Error: 31.5 → 0.4
Time: 17.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0370517283930400648239711358655767980963:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - \sqrt[3]{\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)\right)}}{x \cdot \left(1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0370517283930400648239711358655767980963:\\
\;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - \sqrt[3]{\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)\right)}}{x \cdot \left(1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x\right)}\\

\mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right) + \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r1507935 = 1.0;
        double r1507936 = x;
        double r1507937 = cos(r1507936);
        double r1507938 = r1507935 - r1507937;
        double r1507939 = r1507936 * r1507936;
        double r1507940 = r1507938 / r1507939;
        return r1507940;
}
↓
double f(double x) {
        double r1507941 = x;
        double r1507942 = -0.037051728393040065;
        bool r1507943 = r1507941 <= r1507942;
        double r1507944 = 1.0;
        double r1507945 = r1507944 / r1507941;
        double r1507946 = 1.0;
        double r1507947 = 3.0;
        double r1507948 = pow(r1507946, r1507947);
        double r1507949 = cos(r1507941);
        double r1507950 = r1507949 * r1507949;
        double r1507951 = r1507950 * r1507949;
        double r1507952 = r1507951 * r1507951;
        double r1507953 = r1507951 * r1507952;
        double r1507954 = cbrt(r1507953);
        double r1507955 = r1507948 - r1507954;
        double r1507956 = r1507946 * r1507946;
        double r1507957 = r1507946 + r1507949;
        double r1507958 = r1507957 * r1507949;
        double r1507959 = r1507956 + r1507958;
        double r1507960 = r1507941 * r1507959;
        double r1507961 = r1507955 / r1507960;
        double r1507962 = r1507945 * r1507961;
        double r1507963 = 0.033302914266201536;
        bool r1507964 = r1507941 <= r1507963;
        double r1507965 = r1507941 * r1507941;
        double r1507966 = 0.001388888888888889;
        double r1507967 = r1507965 * r1507966;
        double r1507968 = -0.041666666666666664;
        double r1507969 = r1507967 + r1507968;
        double r1507970 = r1507965 * r1507969;
        double r1507971 = 0.5;
        double r1507972 = r1507970 + r1507971;
        double r1507973 = r1507946 - r1507949;
        double r1507974 = r1507973 / r1507965;
        double r1507975 = r1507964 ? r1507972 : r1507974;
        double r1507976 = r1507943 ? r1507962 : r1507975;
        return r1507976;
}

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 < -0.037051728393040065

    1. Initial program 1.3

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

    3. Applied *-un-lft-identity1.3

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

    4. Applied times-frac0.5

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

    6. Applied flip3--0.6

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

    7. Applied associate-/l/0.6

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

    8. Simplified0.6

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

    10. Applied add-cbrt-cube0.6

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

    11. Simplified0.6

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

    if -0.037051728393040065 < x < 0.033302914266201536

    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 0.033302914266201536 < x

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.5

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

    6. Applied *-un-lft-identity0.5

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

    7. Applied associate-*l*0.5

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

    8. Simplified1.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0370517283930400648239711358655767980963:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - \sqrt[3]{\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)\right)}}{x \cdot \left(1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x\right)}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{-1}{24}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))