Average Error: 31.3 → 0.4
Time: 13.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02898551851456166936205072204302268801257:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{-1}{24} + \frac{1}{720} \cdot \left(x \cdot x\right)\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02898551851456166936205072204302268801257:\\
\;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\

\mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{-1}{24} + \frac{1}{720} \cdot \left(x \cdot x\right)\right) + \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r22058 = 1.0;
        double r22059 = x;
        double r22060 = cos(r22059);
        double r22061 = r22058 - r22060;
        double r22062 = r22059 * r22059;
        double r22063 = r22061 / r22062;
        return r22063;
}
↓
double f(double x) {
        double r22064 = x;
        double r22065 = -0.02898551851456167;
        bool r22066 = r22064 <= r22065;
        double r22067 = 1.0;
        double r22068 = 1.0;
        double r22069 = cos(r22064);
        double r22070 = r22068 - r22069;
        double r22071 = r22064 / r22070;
        double r22072 = r22067 / r22071;
        double r22073 = r22072 / r22064;
        double r22074 = 0.02891102314655495;
        bool r22075 = r22064 <= r22074;
        double r22076 = r22064 * r22064;
        double r22077 = -0.041666666666666664;
        double r22078 = 0.001388888888888889;
        double r22079 = r22078 * r22076;
        double r22080 = r22077 + r22079;
        double r22081 = r22076 * r22080;
        double r22082 = 0.5;
        double r22083 = r22081 + r22082;
        double r22084 = r22070 / r22064;
        double r22085 = r22064 / r22084;
        double r22086 = r22067 / r22085;
        double r22087 = r22075 ? r22083 : r22086;
        double r22088 = r22066 ? r22073 : r22087;
        return r22088;
}

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 < -0.02898551851456167

    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 clear-num0.5

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

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

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

    if -0.02898551851456167 < x < 0.02891102314655495

    1. Initial program 62.2

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

    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 0.02891102314655495 < x

    1. Initial program 0.9

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

    3. Applied clear-num0.9

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

    4. Simplified0.9

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02898551851456166936205072204302268801257:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{-1}{24} + \frac{1}{720} \cdot \left(x \cdot x\right)\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))