Average Error: 30.7 → 0.3
Time: 9.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03116229552204852:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1\right) \cdot x}}{x}\\ \mathbf{elif}\;x \le 0.032272014734577137:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} \cdot {1}^{3} - {\left(\cos x\right)}^{3} \cdot {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03116229552204852:\\
\;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1\right) \cdot x}}{x}\\

\mathbf{elif}\;x \le 0.032272014734577137:\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{1}^{3} \cdot {1}^{3} - {\left(\cos x\right)}^{3} \cdot {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}}{x}\\

\end{array}
double f(double x) {
        double r143 = 1.0;
        double r144 = x;
        double r145 = cos(r144);
        double r146 = r143 - r145;
        double r147 = r144 * r144;
        double r148 = r146 / r147;
        return r148;
}
↓
double f(double x) {
        double r149 = x;
        double r150 = -0.031162295522048522;
        bool r151 = r149 <= r150;
        double r152 = 1.0;
        double r153 = 3.0;
        double r154 = pow(r152, r153);
        double r155 = cos(r149);
        double r156 = pow(r155, r153);
        double r157 = r154 - r156;
        double r158 = exp(r157);
        double r159 = log(r158);
        double r160 = r155 + r152;
        double r161 = cbrt(r160);
        double r162 = r161 * r161;
        double r163 = r162 * r161;
        double r164 = r155 * r163;
        double r165 = r152 * r152;
        double r166 = r164 + r165;
        double r167 = r166 * r149;
        double r168 = r159 / r167;
        double r169 = r168 / r149;
        double r170 = 0.03227201473457714;
        bool r171 = r149 <= r170;
        double r172 = 0.5;
        double r173 = r172 * r149;
        double r174 = 0.001388888888888889;
        double r175 = 5.0;
        double r176 = pow(r149, r175);
        double r177 = r174 * r176;
        double r178 = r173 + r177;
        double r179 = 0.041666666666666664;
        double r180 = pow(r149, r153);
        double r181 = r179 * r180;
        double r182 = r178 - r181;
        double r183 = r182 / r149;
        double r184 = r154 * r154;
        double r185 = r156 * r156;
        double r186 = r184 - r185;
        double r187 = r155 * r160;
        double r188 = r187 + r165;
        double r189 = r188 * r149;
        double r190 = r154 + r156;
        double r191 = r189 * r190;
        double r192 = r186 / r191;
        double r193 = r192 / r149;
        double r194 = r171 ? r183 : r193;
        double r195 = r151 ? r169 : r194;
        return r195;
}

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 < -0.031162295522048522

    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 flip3--0.5

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

    6. Applied associate-/l/0.5

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

    7. Simplified0.5

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

    9. Applied add-log-exp0.5

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

    10. Applied add-log-exp0.5

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

    11. Applied diff-log0.6

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

    12. Simplified0.5

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

    14. Applied add-cube-cbrt0.6

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

    if -0.031162295522048522 < x < 0.03227201473457714

    1. Initial program 62.2

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

    3. Applied associate-/r*61.3

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

    4. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}}{x}\]

    if 0.03227201473457714 < x

    1. Initial program 1.3

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 flip3--0.5

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

    6. Applied associate-/l/0.5

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

    7. Simplified0.5

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

    9. Applied flip--0.6

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

    10. Applied associate-/l/0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03116229552204852:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\left(\sqrt[3]{\cos x + 1} \cdot \sqrt[3]{\cos x + 1}\right) \cdot \sqrt[3]{\cos x + 1}\right) + 1 \cdot 1\right) \cdot x}}{x}\\ \mathbf{elif}\;x \le 0.032272014734577137:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot x + \frac{1}{720} \cdot {x}^{5}\right) - \frac{1}{24} \cdot {x}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} \cdot {1}^{3} - {\left(\cos x\right)}^{3} \cdot {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))