Average Error: 31.4 → 0.3
Time: 4.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\
\;\;\;\;\frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\

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

\end{array}
double f(double x) {
        double r39438 = 1.0;
        double r39439 = x;
        double r39440 = cos(r39439);
        double r39441 = r39438 - r39440;
        double r39442 = r39439 * r39439;
        double r39443 = r39441 / r39442;
        return r39443;
}
↓
double f(double x) {
        double r39444 = x;
        double r39445 = -0.03302109030086143;
        bool r39446 = r39444 <= r39445;
        double r39447 = 0.028129396018266136;
        bool r39448 = r39444 <= r39447;
        double r39449 = !r39448;
        bool r39450 = r39446 || r39449;
        double r39451 = 1.0;
        double r39452 = cos(r39444);
        double r39453 = r39451 - r39452;
        double r39454 = r39453 * r39453;
        double r39455 = cbrt(r39454);
        double r39456 = cbrt(r39453);
        double r39457 = r39455 * r39456;
        double r39458 = 3.0;
        double r39459 = pow(r39457, r39458);
        double r39460 = cbrt(r39459);
        double r39461 = sqrt(r39460);
        double r39462 = r39461 / r39444;
        double r39463 = sqrt(r39453);
        double r39464 = r39463 / r39444;
        double r39465 = r39462 * r39464;
        double r39466 = 0.001388888888888889;
        double r39467 = 4.0;
        double r39468 = pow(r39444, r39467);
        double r39469 = r39466 * r39468;
        double r39470 = 0.5;
        double r39471 = r39469 + r39470;
        double r39472 = 0.041666666666666664;
        double r39473 = 2.0;
        double r39474 = pow(r39444, r39473);
        double r39475 = r39472 * r39474;
        double r39476 = r39471 - r39475;
        double r39477 = r39450 ? r39465 : r39476;
        return r39477;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03302109030086143 or 0.028129396018266136 < x

    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 add-cbrt-cube0.6

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

    7. Simplified0.6

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

    9. Applied add-cube-cbrt0.7

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

    10. Simplified0.6

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

    if -0.03302109030086143 < x < 0.028129396018266136

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033021090300861429 \lor \neg \left(x \le 0.0281293960182661361\right):\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{{\left(\sqrt[3]{\left(1 - \cos x\right) \cdot \left(1 - \cos x\right)} \cdot \sqrt[3]{1 - \cos x}\right)}^{3}}}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))