Average Error: 31.5 → 0.6
Time: 11.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03415862644372414:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}} \cdot \sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}}{x} \cdot \frac{\sqrt[3]{\sqrt{1 - \cos x}} \cdot \sqrt[3]{\sqrt{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.034763904894879627:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{2}} - \frac{\cos x}{{x}^{2}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03415862644372414:\\
\;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}} \cdot \sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}}{x} \cdot \frac{\sqrt[3]{\sqrt{1 - \cos x}} \cdot \sqrt[3]{\sqrt{1 - \cos x}}}{x}\\

\mathbf{elif}\;x \le 0.034763904894879627:\\
\;\;\;\;\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\

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

\end{array}
double f(double x) {
        double r36649 = 1.0;
        double r36650 = x;
        double r36651 = cos(r36650);
        double r36652 = r36649 - r36651;
        double r36653 = r36650 * r36650;
        double r36654 = r36652 / r36653;
        return r36654;
}
↓
double f(double x) {
        double r36655 = x;
        double r36656 = -0.03415862644372414;
        bool r36657 = r36655 <= r36656;
        double r36658 = 1.0;
        double r36659 = cos(r36655);
        double r36660 = r36658 - r36659;
        double r36661 = cbrt(r36660);
        double r36662 = r36661 * r36661;
        double r36663 = r36662 * r36661;
        double r36664 = cbrt(r36663);
        double r36665 = 3.0;
        double r36666 = pow(r36658, r36665);
        double r36667 = pow(r36659, r36665);
        double r36668 = r36666 - r36667;
        double r36669 = r36658 + r36659;
        double r36670 = r36659 * r36669;
        double r36671 = fma(r36658, r36658, r36670);
        double r36672 = r36668 / r36671;
        double r36673 = cbrt(r36672);
        double r36674 = r36664 * r36673;
        double r36675 = r36674 / r36655;
        double r36676 = sqrt(r36660);
        double r36677 = cbrt(r36676);
        double r36678 = r36677 * r36677;
        double r36679 = r36678 / r36655;
        double r36680 = r36675 * r36679;
        double r36681 = 0.03476390489487963;
        bool r36682 = r36655 <= r36681;
        double r36683 = 2.0;
        double r36684 = pow(r36655, r36683);
        double r36685 = -0.041666666666666664;
        double r36686 = 0.001388888888888889;
        double r36687 = 4.0;
        double r36688 = pow(r36655, r36687);
        double r36689 = 0.5;
        double r36690 = fma(r36686, r36688, r36689);
        double r36691 = fma(r36684, r36685, r36690);
        double r36692 = r36658 / r36684;
        double r36693 = r36659 / r36684;
        double r36694 = r36692 - r36693;
        double r36695 = r36682 ? r36691 : r36694;
        double r36696 = r36657 ? r36680 : r36695;
        return r36696;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03415862644372414

    1. Initial program 1.1

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

    3. Applied add-cube-cbrt1.5

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

    4. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.8

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

    7. Applied cbrt-prod0.9

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

    9. Applied add-cube-cbrt0.9

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

    11. Applied flip3--0.9

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

    12. Simplified0.9

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

    if -0.03415862644372414 < x < 0.03476390489487963

    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

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

    if 0.03476390489487963 < x

    1. Initial program 1.2

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

    3. Applied div-sub1.3

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

    4. Simplified1.3

      \[\leadsto \color{blue}{\frac{1}{{x}^{2}}} - \frac{\cos x}{x \cdot x}\]

    5. Simplified1.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03415862644372414:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}} \cdot \sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}}{x} \cdot \frac{\sqrt[3]{\sqrt{1 - \cos x}} \cdot \sqrt[3]{\sqrt{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.034763904894879627:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{2}} - \frac{\cos x}{{x}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))