Average Error: 31.3 → 0.3
Time: 4.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02958908839606369869934532346178457373753:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{{1}^{3} - {\left(\sqrt[3]{\cos x}\right)}^{3} \cdot {\left(\cos x\right)}^{2}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \mathbf{elif}\;x \le 0.02287413456233419392638062106470897560939:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\frac{{1}^{6} - {\left(\cos x\right)}^{6}}{{1}^{3} + {\left(\cos x\right)}^{3}}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02958908839606369869934532346178457373753:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{{1}^{3} - {\left(\sqrt[3]{\cos x}\right)}^{3} \cdot {\left(\cos x\right)}^{2}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\

\mathbf{elif}\;x \le 0.02287413456233419392638062106470897560939:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\frac{{1}^{6} - {\left(\cos x\right)}^{6}}{{1}^{3} + {\left(\cos x\right)}^{3}}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\

\end{array}
double f(double x) {
        double r22092 = 1.0;
        double r22093 = x;
        double r22094 = cos(r22093);
        double r22095 = r22092 - r22094;
        double r22096 = r22093 * r22093;
        double r22097 = r22095 / r22096;
        return r22097;
}
↓
double f(double x) {
        double r22098 = x;
        double r22099 = -0.0295890883960637;
        bool r22100 = r22098 <= r22099;
        double r22101 = 1.0;
        double r22102 = cos(r22098);
        double r22103 = r22101 - r22102;
        double r22104 = sqrt(r22103);
        double r22105 = r22104 / r22098;
        double r22106 = 3.0;
        double r22107 = pow(r22101, r22106);
        double r22108 = cbrt(r22102);
        double r22109 = pow(r22108, r22106);
        double r22110 = 2.0;
        double r22111 = pow(r22102, r22110);
        double r22112 = r22109 * r22111;
        double r22113 = r22107 - r22112;
        double r22114 = sqrt(r22113);
        double r22115 = r22101 * r22101;
        double r22116 = r22102 * r22102;
        double r22117 = r22101 * r22102;
        double r22118 = r22116 + r22117;
        double r22119 = r22115 + r22118;
        double r22120 = sqrt(r22119);
        double r22121 = r22098 * r22120;
        double r22122 = r22114 / r22121;
        double r22123 = r22105 * r22122;
        double r22124 = 0.022874134562334194;
        bool r22125 = r22098 <= r22124;
        double r22126 = 0.001388888888888889;
        double r22127 = 4.0;
        double r22128 = pow(r22098, r22127);
        double r22129 = r22126 * r22128;
        double r22130 = 0.5;
        double r22131 = r22129 + r22130;
        double r22132 = 0.041666666666666664;
        double r22133 = pow(r22098, r22110);
        double r22134 = r22132 * r22133;
        double r22135 = r22131 - r22134;
        double r22136 = 6.0;
        double r22137 = pow(r22101, r22136);
        double r22138 = pow(r22102, r22136);
        double r22139 = r22137 - r22138;
        double r22140 = pow(r22102, r22106);
        double r22141 = r22107 + r22140;
        double r22142 = r22139 / r22141;
        double r22143 = sqrt(r22142);
        double r22144 = r22143 / r22121;
        double r22145 = r22105 * r22144;
        double r22146 = r22125 ? r22135 : r22145;
        double r22147 = r22100 ? r22123 : r22146;
        return r22147;
}

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 < -0.0295890883960637

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied times-frac0.6

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

    6. Applied flip3--0.6

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

    7. Applied sqrt-div0.6

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

    8. Applied associate-/l/0.5

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

    10. Applied add-log-exp0.5

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

    12. Applied add-cube-cbrt0.7

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

    13. Applied unpow-prod-down0.8

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

    14. Applied exp-prod0.8

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

    15. Applied log-pow0.7

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

    16. Simplified0.6

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

    if -0.0295890883960637 < x < 0.022874134562334194

    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 0.022874134562334194 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied times-frac0.6

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

    6. Applied flip3--0.5

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

    7. Applied sqrt-div0.5

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

    8. Applied associate-/l/0.5

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

    10. Applied flip--0.6

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

    11. Simplified0.6

      \[\leadsto \frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\frac{\color{blue}{{1}^{6} - {\left(\cos x\right)}^{6}}}{{1}^{3} + {\left(\cos x\right)}^{3}}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02958908839606369869934532346178457373753:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{{1}^{3} - {\left(\sqrt[3]{\cos x}\right)}^{3} \cdot {\left(\cos x\right)}^{2}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \mathbf{elif}\;x \le 0.02287413456233419392638062106470897560939:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\frac{{1}^{6} - {\left(\cos x\right)}^{6}}{{1}^{3} + {\left(\cos x\right)}^{3}}}}{x \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))