Average Error: 31.6 → 0.4
Time: 4.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.037855314436600819:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right) \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.0330388474946485614:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.037855314436600819:\\
\;\;\;\;\frac{\log \left(e^{1 - \cos x}\right) \cdot \frac{1}{x}}{x}\\

\mathbf{elif}\;x \le 0.0330388474946485614:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\

\end{array}
double f(double x) {
        double r28125 = 1.0;
        double r28126 = x;
        double r28127 = cos(r28126);
        double r28128 = r28125 - r28127;
        double r28129 = r28126 * r28126;
        double r28130 = r28128 / r28129;
        return r28130;
}
↓
double f(double x) {
        double r28131 = x;
        double r28132 = -0.03785531443660082;
        bool r28133 = r28131 <= r28132;
        double r28134 = 1.0;
        double r28135 = cos(r28131);
        double r28136 = r28134 - r28135;
        double r28137 = exp(r28136);
        double r28138 = log(r28137);
        double r28139 = 1.0;
        double r28140 = r28139 / r28131;
        double r28141 = r28138 * r28140;
        double r28142 = r28141 / r28131;
        double r28143 = 0.03303884749464856;
        bool r28144 = r28131 <= r28143;
        double r28145 = 0.001388888888888889;
        double r28146 = 4.0;
        double r28147 = pow(r28131, r28146);
        double r28148 = r28145 * r28147;
        double r28149 = 0.5;
        double r28150 = r28148 + r28149;
        double r28151 = 0.041666666666666664;
        double r28152 = 2.0;
        double r28153 = pow(r28131, r28152);
        double r28154 = r28151 * r28153;
        double r28155 = r28150 - r28154;
        double r28156 = r28131 * r28131;
        double r28157 = r28138 / r28156;
        double r28158 = r28144 ? r28155 : r28157;
        double r28159 = r28133 ? r28142 : r28158;
        return r28159;
}

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 < -0.03785531443660082

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 add-log-exp0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied diff-log0.7

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}}{x}\]

    8. Simplified0.6

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

    10. Applied div-inv0.6

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

    if -0.03785531443660082 < x < 0.03303884749464856

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    if 0.03303884749464856 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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-log-exp0.5

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

    6. Applied add-log-exp0.5

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

    7. Applied diff-log0.6

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}}{x}\]

    8. Simplified0.5

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

    10. Applied div-inv0.6

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

    11. Applied associate-/l*1.0

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

    12. Simplified1.0

      \[\leadsto \frac{\log \left(e^{1 - \cos x}\right)}{\color{blue}{x \cdot x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.037855314436600819:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right) \cdot \frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.0330388474946485614:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))