Average Error: 31.8 → 0.4
Time: 5.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03422954134275022625155671107677335385233 \lor \neg \left(x \le 0.03575238836244829659927191300994309131056\right):\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \left|\sqrt[3]{1 - \cos x}\right|}{x} \cdot \frac{\sqrt{\sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right)}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03422954134275022625155671107677335385233 \lor \neg \left(x \le 0.03575238836244829659927191300994309131056\right):\\
\;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \left|\sqrt[3]{1 - \cos x}\right|}{x} \cdot \frac{\sqrt{\sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right)}}}}{x}\\

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

\end{array}
double f(double x) {
        double r36368 = 1.0;
        double r36369 = x;
        double r36370 = cos(r36369);
        double r36371 = r36368 - r36370;
        double r36372 = r36369 * r36369;
        double r36373 = r36371 / r36372;
        return r36373;
}
↓
double f(double x) {
        double r36374 = x;
        double r36375 = -0.034229541342750226;
        bool r36376 = r36374 <= r36375;
        double r36377 = 0.0357523883624483;
        bool r36378 = r36374 <= r36377;
        double r36379 = !r36378;
        bool r36380 = r36376 || r36379;
        double r36381 = 1.0;
        double r36382 = cos(r36374);
        double r36383 = r36381 - r36382;
        double r36384 = exp(r36383);
        double r36385 = log(r36384);
        double r36386 = sqrt(r36385);
        double r36387 = cbrt(r36383);
        double r36388 = fabs(r36387);
        double r36389 = r36386 * r36388;
        double r36390 = r36389 / r36374;
        double r36391 = 3.0;
        double r36392 = pow(r36381, r36391);
        double r36393 = pow(r36382, r36391);
        double r36394 = r36392 - r36393;
        double r36395 = r36382 + r36381;
        double r36396 = r36382 * r36395;
        double r36397 = fma(r36381, r36381, r36396);
        double r36398 = r36394 / r36397;
        double r36399 = cbrt(r36398);
        double r36400 = sqrt(r36399);
        double r36401 = r36400 / r36374;
        double r36402 = r36390 * r36401;
        double r36403 = 4.0;
        double r36404 = pow(r36374, r36403);
        double r36405 = 0.001388888888888889;
        double r36406 = 0.5;
        double r36407 = 0.041666666666666664;
        double r36408 = 2.0;
        double r36409 = pow(r36374, r36408);
        double r36410 = r36407 * r36409;
        double r36411 = r36406 - r36410;
        double r36412 = fma(r36404, r36405, r36411);
        double r36413 = r36380 ? r36402 : r36412;
        return r36413;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.034229541342750226 or 0.0357523883624483 < x

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.3

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

    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-cube-cbrt0.7

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

    8. Applied sqrt-prod0.7

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

    9. Applied times-frac0.7

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

    10. Applied associate-*r*0.7

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

    11. Simplified0.7

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

    13. Applied add-log-exp0.7

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

    14. Applied add-log-exp0.7

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

    15. Applied diff-log0.8

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

    16. Simplified0.7

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

    18. Applied flip3--0.7

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

    19. Simplified0.7

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

    if -0.034229541342750226 < x < 0.0357523883624483

    1. Initial program 62.1

      \[\frac{1 - \cos x}{x \cdot x}\]

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03422954134275022625155671107677335385233 \lor \neg \left(x \le 0.03575238836244829659927191300994309131056\right):\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \left|\sqrt[3]{1 - \cos x}\right|}{x} \cdot \frac{\sqrt{\sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right)}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))