Average Error: 31.3 → 0.5
Time: 18.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03472819889887164557240595286202733404934:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03472819889887164557240595286202733404934:\\
\;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\
\;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right)\\

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

\end{array}
double f(double x) {
        double r2064714 = 1.0;
        double r2064715 = x;
        double r2064716 = cos(r2064715);
        double r2064717 = r2064714 - r2064716;
        double r2064718 = r2064715 * r2064715;
        double r2064719 = r2064717 / r2064718;
        return r2064719;
}
↓
double f(double x) {
        double r2064720 = x;
        double r2064721 = -0.034728198898871646;
        bool r2064722 = r2064720 <= r2064721;
        double r2064723 = 1.0;
        double r2064724 = r2064723 / r2064720;
        double r2064725 = cos(r2064720);
        double r2064726 = r2064725 / r2064720;
        double r2064727 = r2064724 - r2064726;
        double r2064728 = r2064727 / r2064720;
        double r2064729 = 0.036569739872455896;
        bool r2064730 = r2064720 <= r2064729;
        double r2064731 = 0.5;
        double r2064732 = r2064720 * r2064720;
        double r2064733 = 0.001388888888888889;
        double r2064734 = r2064733 * r2064732;
        double r2064735 = 0.041666666666666664;
        double r2064736 = r2064734 - r2064735;
        double r2064737 = r2064732 * r2064736;
        double r2064738 = r2064731 + r2064737;
        double r2064739 = r2064723 / r2064732;
        double r2064740 = r2064725 / r2064732;
        double r2064741 = r2064739 - r2064740;
        double r2064742 = r2064730 ? r2064738 : r2064741;
        double r2064743 = r2064722 ? r2064728 : r2064742;
        return r2064743;
}

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 < -0.034728198898871646

    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 associate-*r/0.6

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

    7. Simplified0.5

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

    9. Applied div-sub0.6

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

    if -0.034728198898871646 < x < 0.036569739872455896

    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. Simplified0.0

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

    if 0.036569739872455896 < x

    1. Initial program 1.1

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

    3. Applied div-sub1.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03472819889887164557240595286202733404934:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\ \;\;\;\;\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))