Average Error: 31.5 → 0.4
Time: 14.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03405516779529376564239484537210955750197:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\frac{\frac{1}{720} \cdot {x}^{5} + \left(x \cdot \frac{1}{2} - {x}^{3} \cdot \frac{1}{24}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03405516779529376564239484537210955750197:\\
\;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\
\;\;\;\;\frac{\frac{1}{720} \cdot {x}^{5} + \left(x \cdot \frac{1}{2} - {x}^{3} \cdot \frac{1}{24}\right)}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\

\end{array}
double f(double x) {
        double r31983 = 1.0;
        double r31984 = x;
        double r31985 = cos(r31984);
        double r31986 = r31983 - r31985;
        double r31987 = r31984 * r31984;
        double r31988 = r31986 / r31987;
        return r31988;
}
↓
double f(double x) {
        double r31989 = x;
        double r31990 = -0.034055167795293766;
        bool r31991 = r31989 <= r31990;
        double r31992 = 1.0;
        double r31993 = r31992 / r31989;
        double r31994 = 1.0;
        double r31995 = cos(r31989);
        double r31996 = r31994 - r31995;
        double r31997 = r31989 / r31996;
        double r31998 = r31993 / r31997;
        double r31999 = 0.033302914266201536;
        bool r32000 = r31989 <= r31999;
        double r32001 = 0.001388888888888889;
        double r32002 = 5.0;
        double r32003 = pow(r31989, r32002);
        double r32004 = r32001 * r32003;
        double r32005 = 0.5;
        double r32006 = r31989 * r32005;
        double r32007 = 3.0;
        double r32008 = pow(r31989, r32007);
        double r32009 = 0.041666666666666664;
        double r32010 = r32008 * r32009;
        double r32011 = r32006 - r32010;
        double r32012 = r32004 + r32011;
        double r32013 = r32012 / r31989;
        double r32014 = 2.0;
        double r32015 = pow(r31989, r32014);
        double r32016 = r31996 / r32015;
        double r32017 = r32000 ? r32013 : r32016;
        double r32018 = r31991 ? r31998 : r32017;
        return r32018;
}

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 < -0.034055167795293766

    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 div-sub0.6

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

    7. Applied div-inv0.6

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

    8. Applied div-inv0.6

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

    9. Applied distribute-rgt-out--0.5

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

    10. Applied associate-/l*0.5

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

    if -0.034055167795293766 < x < 0.033302914266201536

    1. Initial program 62.3

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

    3. Applied associate-/r*61.3

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

    5. Applied div-sub61.3

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

    6. Taylor expanded around 0 0.0

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

    7. Simplified0.0

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

    if 0.033302914266201536 < x

    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 div-sub0.6

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

    7. Applied sub-div0.5

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

    8. Applied associate-/l/1.1

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

    9. Simplified1.1

      \[\leadsto \frac{1 - \cos x}{\color{blue}{{x}^{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03405516779529376564239484537210955750197:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\frac{\frac{1}{720} \cdot {x}^{5} + \left(x \cdot \frac{1}{2} - {x}^{3} \cdot \frac{1}{24}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{{x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))