Average Error: 31.5 → 0.4
Time: 4.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0301372104399350062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \le 0.028183382960144138:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{\mathsf{fma}\left(-{x}^{6}, \frac{1}{13824}, \frac{1}{8}\right)}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{576}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0301372104399350062:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}\\

\mathbf{elif}\;x \le 0.028183382960144138:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{\mathsf{fma}\left(-{x}^{6}, \frac{1}{13824}, \frac{1}{8}\right)}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{576}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\

\end{array}
double f(double x) {
        double r27597 = 1.0;
        double r27598 = x;
        double r27599 = cos(r27598);
        double r27600 = r27597 - r27599;
        double r27601 = r27598 * r27598;
        double r27602 = r27600 / r27601;
        return r27602;
}
↓
double f(double x) {
        double r27603 = x;
        double r27604 = -0.030137210439935006;
        bool r27605 = r27603 <= r27604;
        double r27606 = 1.0;
        double r27607 = r27606 / r27603;
        double r27608 = 1.0;
        double r27609 = r27608 * r27608;
        double r27610 = cos(r27603);
        double r27611 = r27610 * r27610;
        double r27612 = r27609 - r27611;
        double r27613 = r27608 + r27610;
        double r27614 = r27603 * r27613;
        double r27615 = r27612 / r27614;
        double r27616 = r27607 * r27615;
        double r27617 = 0.028183382960144138;
        bool r27618 = r27603 <= r27617;
        double r27619 = 4.0;
        double r27620 = pow(r27603, r27619);
        double r27621 = 0.001388888888888889;
        double r27622 = 6.0;
        double r27623 = pow(r27603, r27622);
        double r27624 = -r27623;
        double r27625 = 7.233796296296296e-05;
        double r27626 = 0.125;
        double r27627 = fma(r27624, r27625, r27626);
        double r27628 = 2.0;
        double r27629 = pow(r27603, r27628);
        double r27630 = 0.001736111111111111;
        double r27631 = 0.020833333333333332;
        double r27632 = fma(r27629, r27630, r27631);
        double r27633 = 0.25;
        double r27634 = fma(r27632, r27629, r27633);
        double r27635 = r27627 / r27634;
        double r27636 = fma(r27620, r27621, r27635);
        double r27637 = r27608 / r27603;
        double r27638 = r27610 / r27603;
        double r27639 = r27637 - r27638;
        double r27640 = r27607 * r27639;
        double r27641 = r27618 ? r27636 : r27640;
        double r27642 = r27605 ? r27616 : r27641;
        return r27642;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.030137210439935006

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.5

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

    6. Applied flip--0.7

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

    7. Applied associate-/l/0.7

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

    if -0.030137210439935006 < x < 0.028183382960144138

    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. Simplified0.0

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

    5. Applied flip3--0.0

      \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{1}{720}, \color{blue}{\frac{{\frac{1}{2}}^{3} - {\left(\frac{1}{24} \cdot {x}^{2}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right) + \frac{1}{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}}\right)\]

    6. Simplified0.0

      \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{\color{blue}{\mathsf{fma}\left(-{x}^{6}, \frac{1}{13824}, \frac{1}{8}\right)}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right) + \frac{1}{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\]

    7. Simplified0.0

      \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{\mathsf{fma}\left(-{x}^{6}, \frac{1}{13824}, \frac{1}{8}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{576}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right)}}\right)\]

    if 0.028183382960144138 < x

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied div-sub0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0301372104399350062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 \cdot 1 - \cos x \cdot \cos x}{x \cdot \left(1 + \cos x\right)}\\ \mathbf{elif}\;x \le 0.028183382960144138:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{\mathsf{fma}\left(-{x}^{6}, \frac{1}{13824}, \frac{1}{8}\right)}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{576}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))