Average Error: 32.0 → 0.2
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031030980254842244 \lor \neg \left(x \le 0.0254535513975123508\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031030980254842244 \lor \neg \left(x \le 0.0254535513975123508\right):\\
\;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\\

\end{array}
double f(double x) {
        double r35534 = 1.0;
        double r35535 = x;
        double r35536 = cos(r35535);
        double r35537 = r35534 - r35536;
        double r35538 = r35535 * r35535;
        double r35539 = r35537 / r35538;
        return r35539;
}
↓
double f(double x) {
        double r35540 = x;
        double r35541 = -0.031030980254842244;
        bool r35542 = r35540 <= r35541;
        double r35543 = 0.02545355139751235;
        bool r35544 = r35540 <= r35543;
        double r35545 = !r35544;
        bool r35546 = r35542 || r35545;
        double r35547 = 1.0;
        double r35548 = cos(r35540);
        double r35549 = r35547 - r35548;
        double r35550 = r35549 / r35540;
        double r35551 = 1.0;
        double r35552 = r35551 / r35540;
        double r35553 = r35550 * r35552;
        double r35554 = 0.001736111111111111;
        double r35555 = -r35554;
        double r35556 = 4.0;
        double r35557 = pow(r35540, r35556);
        double r35558 = r35555 * r35557;
        double r35559 = 0.001388888888888889;
        double r35560 = r35559 * r35557;
        double r35561 = 0.5;
        double r35562 = r35560 + r35561;
        double r35563 = r35562 * r35562;
        double r35564 = r35558 + r35563;
        double r35565 = 0.041666666666666664;
        double r35566 = 2.0;
        double r35567 = pow(r35540, r35566);
        double r35568 = r35565 * r35567;
        double r35569 = r35562 + r35568;
        double r35570 = r35564 / r35569;
        double r35571 = r35546 ? r35553 : r35570;
        return r35571;
}

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 < -0.031030980254842244 or 0.02545355139751235 < 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-inv0.5

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

    if -0.031030980254842244 < x < 0.02545355139751235

    1. Initial program 62.4

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
    3. Using strategy rm

    4. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}}\]

    5. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031030980254842244 \lor \neg \left(x \le 0.0254535513975123508\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))