Average Error: 31.3 → 0.5
Time: 22.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02966430694204434670435688303768984042108:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \sqrt{\sqrt{1 - \cos x}}}{x} \cdot \frac{\sqrt{\sqrt{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02966430694204434670435688303768984042108:\\
\;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \sqrt{\sqrt{1 - \cos x}}}{x} \cdot \frac{\sqrt{\sqrt{1 - \cos x}}}{x}\\

\mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\

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

\end{array}
double f(double x) {
        double r2287637 = 1.0;
        double r2287638 = x;
        double r2287639 = cos(r2287638);
        double r2287640 = r2287637 - r2287639;
        double r2287641 = r2287638 * r2287638;
        double r2287642 = r2287640 / r2287641;
        return r2287642;
}
↓
double f(double x) {
        double r2287643 = x;
        double r2287644 = -0.029664306942044347;
        bool r2287645 = r2287643 <= r2287644;
        double r2287646 = 1.0;
        double r2287647 = cos(r2287643);
        double r2287648 = r2287646 - r2287647;
        double r2287649 = exp(r2287648);
        double r2287650 = log(r2287649);
        double r2287651 = sqrt(r2287650);
        double r2287652 = sqrt(r2287648);
        double r2287653 = sqrt(r2287652);
        double r2287654 = r2287651 * r2287653;
        double r2287655 = r2287654 / r2287643;
        double r2287656 = r2287653 / r2287643;
        double r2287657 = r2287655 * r2287656;
        double r2287658 = 0.036569739872455896;
        bool r2287659 = r2287643 <= r2287658;
        double r2287660 = -0.041666666666666664;
        double r2287661 = r2287643 * r2287643;
        double r2287662 = 0.001388888888888889;
        double r2287663 = r2287661 * r2287661;
        double r2287664 = 0.5;
        double r2287665 = fma(r2287662, r2287663, r2287664);
        double r2287666 = fma(r2287660, r2287661, r2287665);
        double r2287667 = r2287646 / r2287661;
        double r2287668 = r2287647 / r2287661;
        double r2287669 = r2287667 - r2287668;
        double r2287670 = r2287659 ? r2287666 : r2287669;
        double r2287671 = r2287645 ? r2287657 : r2287670;
        return r2287671;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.029664306942044347

    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 *-un-lft-identity0.6

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

    7. Applied add-sqr-sqrt0.6

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

    8. Applied sqrt-prod0.7

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

    9. Applied times-frac0.6

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

    10. Applied associate-*r*0.6

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

    11. Simplified0.6

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

    13. Applied add-log-exp0.6

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

    14. Applied add-log-exp0.6

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

    15. Applied diff-log0.7

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

    16. Simplified0.6

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

    if -0.029664306942044347 < x < 0.036569739872455896

    1. Initial program 62.2

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

    3. Applied add-sqr-sqrt62.2

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

    4. Applied times-frac61.3

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

    5. Taylor expanded around 0 0.0

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

    6. Simplified0.0

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

    if 0.036569739872455896 < x

    1. Initial program 1.1

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

    3. Applied div-sub1.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02966430694204434670435688303768984042108:\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \sqrt{\sqrt{1 - \cos x}}}{x} \cdot \frac{\sqrt{\sqrt{1 - \cos x}}}{x}\\ \mathbf{elif}\;x \le 0.03656973987245589613470642120773845817894:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))