Average Error: 31.3 → 0.3
Time: 4.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02680284925783882 \lor \neg \left(x \le 0.0326302713396224922\right):\\ \;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02680284925783882 \lor \neg \left(x \le 0.0326302713396224922\right):\\
\;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{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 r30759 = 1.0;
        double r30760 = x;
        double r30761 = cos(r30760);
        double r30762 = r30759 - r30761;
        double r30763 = r30760 * r30760;
        double r30764 = r30762 / r30763;
        return r30764;
}
↓
double f(double x) {
        double r30765 = x;
        double r30766 = -0.026802849257838823;
        bool r30767 = r30765 <= r30766;
        double r30768 = 0.03263027133962249;
        bool r30769 = r30765 <= r30768;
        double r30770 = !r30769;
        bool r30771 = r30767 || r30770;
        double r30772 = 1.0;
        double r30773 = 3.0;
        double r30774 = pow(r30772, r30773);
        double r30775 = cos(r30765);
        double r30776 = pow(r30775, r30773);
        double r30777 = r30774 - r30776;
        double r30778 = r30775 + r30772;
        double r30779 = r30775 * r30778;
        double r30780 = r30772 * r30772;
        double r30781 = r30779 + r30780;
        double r30782 = r30777 / r30781;
        double r30783 = sqrt(r30782);
        double r30784 = r30783 / r30765;
        double r30785 = r30772 - r30775;
        double r30786 = sqrt(r30785);
        double r30787 = r30786 / r30765;
        double r30788 = r30784 * r30787;
        double r30789 = 0.001736111111111111;
        double r30790 = -r30789;
        double r30791 = 4.0;
        double r30792 = pow(r30765, r30791);
        double r30793 = r30790 * r30792;
        double r30794 = 0.001388888888888889;
        double r30795 = r30794 * r30792;
        double r30796 = 0.5;
        double r30797 = r30795 + r30796;
        double r30798 = r30797 * r30797;
        double r30799 = r30793 + r30798;
        double r30800 = 0.041666666666666664;
        double r30801 = 2.0;
        double r30802 = pow(r30765, r30801);
        double r30803 = r30800 * r30802;
        double r30804 = r30797 + r30803;
        double r30805 = r30799 / r30804;
        double r30806 = r30771 ? r30788 : r30805;
        return r30806;
}

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 < -0.026802849257838823 or 0.03263027133962249 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    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 flip3--0.6

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

    7. Simplified0.6

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

    if -0.026802849257838823 < x < 0.03263027133962249

    1. Initial program 62.3

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

    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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02680284925783882 \lor \neg \left(x \le 0.0326302713396224922\right):\\ \;\;\;\;\frac{\sqrt{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{1}{576}\right) \cdot {x}^{4} + \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right)}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) + \frac{1}{24} \cdot {x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))