Average Error: 32.0 → 0.4
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0286652458901892858:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{{x}^{2} \cdot \mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.0328059247821943858:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \log \left(e^{\frac{1}{24} \cdot {x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0286652458901892858:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{{x}^{2} \cdot \mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\\

\mathbf{elif}\;x \le 0.0328059247821943858:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \log \left(e^{\frac{1}{24} \cdot {x}^{2}}\right)\right)\\

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

\end{array}
double f(double x) {
        double r19632 = 1.0;
        double r19633 = x;
        double r19634 = cos(r19633);
        double r19635 = r19632 - r19634;
        double r19636 = r19633 * r19633;
        double r19637 = r19635 / r19636;
        return r19637;
}
↓
double f(double x) {
        double r19638 = x;
        double r19639 = -0.028665245890189286;
        bool r19640 = r19638 <= r19639;
        double r19641 = 1.0;
        double r19642 = 3.0;
        double r19643 = pow(r19641, r19642);
        double r19644 = cos(r19638);
        double r19645 = pow(r19644, r19642);
        double r19646 = r19643 - r19645;
        double r19647 = 2.0;
        double r19648 = pow(r19638, r19647);
        double r19649 = r19641 + r19644;
        double r19650 = r19641 * r19641;
        double r19651 = fma(r19644, r19649, r19650);
        double r19652 = r19648 * r19651;
        double r19653 = r19646 / r19652;
        double r19654 = 0.032805924782194386;
        bool r19655 = r19638 <= r19654;
        double r19656 = 4.0;
        double r19657 = pow(r19638, r19656);
        double r19658 = 0.001388888888888889;
        double r19659 = 0.5;
        double r19660 = 0.041666666666666664;
        double r19661 = r19660 * r19648;
        double r19662 = exp(r19661);
        double r19663 = log(r19662);
        double r19664 = r19659 - r19663;
        double r19665 = fma(r19657, r19658, r19664);
        double r19666 = r19641 / r19638;
        double r19667 = r19644 / r19638;
        double r19668 = r19666 - r19667;
        double r19669 = r19668 / r19638;
        double r19670 = r19655 ? r19665 : r19669;
        double r19671 = r19640 ? r19653 : r19670;
        return r19671;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.028665245890189286

    1. Initial program 1.2

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

    3. Applied flip3--1.2

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

    4. Applied associate-/l/1.2

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

    5. Simplified1.2

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

    if -0.028665245890189286 < x < 0.032805924782194386

    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 add-log-exp0.0

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

    if 0.032805924782194386 < 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-sub0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0286652458901892858:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{{x}^{2} \cdot \mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.0328059247821943858:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \log \left(e^{\frac{1}{24} \cdot {x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

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