Average Error: 31.4 → 0.3
Time: 4.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x}\\

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

\end{array}
double f(double x) {
        double r23717 = 1.0;
        double r23718 = x;
        double r23719 = cos(r23718);
        double r23720 = r23717 - r23719;
        double r23721 = r23718 * r23718;
        double r23722 = r23720 / r23721;
        return r23722;
}
↓
double f(double x) {
        double r23723 = x;
        double r23724 = -0.03302109030086143;
        bool r23725 = r23723 <= r23724;
        double r23726 = 0.028129396018266136;
        bool r23727 = r23723 <= r23726;
        double r23728 = !r23727;
        bool r23729 = r23725 || r23728;
        double r23730 = 1.0;
        double r23731 = cos(r23723);
        double r23732 = r23730 - r23731;
        double r23733 = sqrt(r23732);
        double r23734 = r23733 / r23723;
        double r23735 = r23732 * r23732;
        double r23736 = cbrt(r23735);
        double r23737 = cbrt(r23732);
        double r23738 = r23736 * r23737;
        double r23739 = 3.0;
        double r23740 = pow(r23738, r23739);
        double r23741 = cbrt(r23740);
        double r23742 = sqrt(r23741);
        double r23743 = r23742 / r23723;
        double r23744 = r23734 * r23743;
        double r23745 = 0.001388888888888889;
        double r23746 = 4.0;
        double r23747 = pow(r23723, r23746);
        double r23748 = r23745 * r23747;
        double r23749 = 0.5;
        double r23750 = r23748 + r23749;
        double r23751 = 0.041666666666666664;
        double r23752 = 2.0;
        double r23753 = pow(r23723, r23752);
        double r23754 = r23751 * r23753;
        double r23755 = r23750 - r23754;
        double r23756 = r23729 ? r23744 : r23755;
        return r23756;
}

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 < -0.03302109030086143 or 0.028129396018266136 < x

    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 add-cbrt-cube0.6

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

    7. Simplified0.6

      \[\leadsto \frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\sqrt[3]{\color{blue}{{\left(1 - \cos x\right)}^{3}}}}}{x}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.7

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

    10. Simplified0.6

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

    if -0.03302109030086143 < x < 0.028129396018266136

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))