Average Error: 31.5 → 0.3
Time: 12.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0319510738025488294500320307633955962956 \lor \neg \left(x \le 0.02827821813838137660068738910013053100556\right):\\ \;\;\;\;\frac{\frac{e^{\log \left(\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)\right) - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0319510738025488294500320307633955962956 \lor \neg \left(x \le 0.02827821813838137660068738910013053100556\right):\\
\;\;\;\;\frac{\frac{e^{\log \left(\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)\right) - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}{x}}{x}\\

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

\end{array}
double f(double x) {
        double r21092 = 1.0;
        double r21093 = x;
        double r21094 = cos(r21093);
        double r21095 = r21092 - r21094;
        double r21096 = r21093 * r21093;
        double r21097 = r21095 / r21096;
        return r21097;
}
↓
double f(double x) {
        double r21098 = x;
        double r21099 = -0.03195107380254883;
        bool r21100 = r21098 <= r21099;
        double r21101 = 0.028278218138381377;
        bool r21102 = r21098 <= r21101;
        double r21103 = !r21102;
        bool r21104 = r21100 || r21103;
        double r21105 = 1.0;
        double r21106 = 3.0;
        double r21107 = pow(r21105, r21106);
        double r21108 = cos(r21098);
        double r21109 = pow(r21108, r21106);
        double r21110 = r21107 - r21109;
        double r21111 = exp(r21110);
        double r21112 = log(r21111);
        double r21113 = log(r21112);
        double r21114 = r21105 * r21105;
        double r21115 = r21108 * r21108;
        double r21116 = r21105 * r21108;
        double r21117 = r21115 + r21116;
        double r21118 = r21114 + r21117;
        double r21119 = log(r21118);
        double r21120 = r21113 - r21119;
        double r21121 = exp(r21120);
        double r21122 = r21121 / r21098;
        double r21123 = r21122 / r21098;
        double r21124 = 0.001388888888888889;
        double r21125 = 4.0;
        double r21126 = pow(r21098, r21125);
        double r21127 = r21124 * r21126;
        double r21128 = 0.5;
        double r21129 = r21127 + r21128;
        double r21130 = 0.041666666666666664;
        double r21131 = 2.0;
        double r21132 = pow(r21098, r21131);
        double r21133 = r21130 * r21132;
        double r21134 = r21129 - r21133;
        double r21135 = r21104 ? r21123 : r21134;
        return r21135;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03195107380254883 or 0.028278218138381377 < x

    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 add-exp-log0.5

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

    7. Applied flip3--0.5

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

    8. Applied log-div0.5

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

    10. Applied add-log-exp0.6

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

    11. Applied add-log-exp0.6

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

    12. Applied diff-log0.6

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

    13. Simplified0.6

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

    if -0.03195107380254883 < x < 0.028278218138381377

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0319510738025488294500320307633955962956 \lor \neg \left(x \le 0.02827821813838137660068738910013053100556\right):\\ \;\;\;\;\frac{\frac{e^{\log \left(\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)\right) - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))