Average Error: 31.2 → 0.4
Time: 26.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\ \mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\

\mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r20093 = 1.0;
        double r20094 = x;
        double r20095 = cos(r20094);
        double r20096 = r20093 - r20095;
        double r20097 = r20094 * r20094;
        double r20098 = r20096 / r20097;
        return r20098;
}
↓
double f(double x) {
        double r20099 = x;
        double r20100 = -0.030846791202208904;
        bool r20101 = r20099 <= r20100;
        double r20102 = 1.0;
        double r20103 = cos(r20099);
        double r20104 = r20102 - r20103;
        double r20105 = sqrt(r20104);
        double r20106 = r20105 / r20099;
        double r20107 = r20099 / r20106;
        double r20108 = r20105 / r20107;
        double r20109 = 0.03981852127604897;
        bool r20110 = r20099 <= r20109;
        double r20111 = 0.001388888888888889;
        double r20112 = 4.0;
        double r20113 = pow(r20099, r20112);
        double r20114 = r20111 * r20113;
        double r20115 = 0.5;
        double r20116 = r20114 + r20115;
        double r20117 = 0.041666666666666664;
        double r20118 = 2.0;
        double r20119 = pow(r20099, r20118);
        double r20120 = r20117 * r20119;
        double r20121 = r20116 - r20120;
        double r20122 = r20102 / r20099;
        double r20123 = r20103 / r20099;
        double r20124 = r20122 - r20123;
        double r20125 = r20124 / r20099;
        double r20126 = r20110 ? r20121 : r20125;
        double r20127 = r20101 ? r20108 : r20126;
        return r20127;
}

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 < -0.030846791202208904

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 *-un-lft-identity0.5

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

    6. Applied add-sqr-sqrt0.6

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

    7. Applied times-frac0.6

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

    8. Applied associate-/l*1.1

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

    if -0.030846791202208904 < x < 0.03981852127604897

    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 0.03981852127604897 < x

    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 div-sub0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03084679120220890427561677427092945436016:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\\ \mathbf{elif}\;x \le 0.03981852127604897095825009500913438387215:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))