Average Error: 31.0 → 0.4
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.029735392258237861:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\ \mathbf{elif}\;x \le 0.033264802814976693:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.029735392258237861:\\
\;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\

\mathbf{elif}\;x \le 0.033264802814976693:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r23130 = 1.0;
        double r23131 = x;
        double r23132 = cos(r23131);
        double r23133 = r23130 - r23132;
        double r23134 = r23131 * r23131;
        double r23135 = r23133 / r23134;
        return r23135;
}
↓
double f(double x) {
        double r23136 = x;
        double r23137 = -0.02973539225823786;
        bool r23138 = r23136 <= r23137;
        double r23139 = 1.0;
        double r23140 = cos(r23136);
        double r23141 = r23139 - r23140;
        double r23142 = 3.0;
        double r23143 = pow(r23141, r23142);
        double r23144 = cbrt(r23143);
        double r23145 = r23144 / r23136;
        double r23146 = r23145 / r23136;
        double r23147 = 0.03326480281497669;
        bool r23148 = r23136 <= r23147;
        double r23149 = 0.001388888888888889;
        double r23150 = 4.0;
        double r23151 = pow(r23136, r23150);
        double r23152 = r23149 * r23151;
        double r23153 = 0.5;
        double r23154 = r23152 + r23153;
        double r23155 = 0.041666666666666664;
        double r23156 = 2.0;
        double r23157 = pow(r23136, r23156);
        double r23158 = r23155 * r23157;
        double r23159 = r23154 - r23158;
        double r23160 = r23136 * r23136;
        double r23161 = r23139 / r23160;
        double r23162 = r23140 / r23160;
        double r23163 = r23161 - r23162;
        double r23164 = r23148 ? r23159 : r23163;
        double r23165 = r23138 ? r23146 : r23164;
        return r23165;
}

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 < -0.02973539225823786

    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 add-cbrt-cube0.6

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

    6. Simplified0.6

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

    if -0.02973539225823786 < x < 0.03326480281497669

    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 0.03326480281497669 < x

    1. Initial program 1.0

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

    3. Applied div-sub1.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029735392258237861:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x}}{x}\\ \mathbf{elif}\;x \le 0.033264802814976693:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))