Average Error: 31.8 → 0.4
Time: 4.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03422954134275022625155671107677335385233 \lor \neg \left(x \le 0.03575238836244829659927191300994309131056\right):\\ \;\;\;\;\frac{\sqrt{\log \left(e^{1 - \cos x}\right)} \cdot \left|\sqrt[3]{1 - \cos x}\right|}{x} \cdot \frac{\sqrt{\sqrt[3]{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03422954134275022625155671107677335385233 \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}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}}{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 r25569 = 1.0;
        double r25570 = x;
        double r25571 = cos(r25570);
        double r25572 = r25569 - r25571;
        double r25573 = r25570 * r25570;
        double r25574 = r25572 / r25573;
        return r25574;
}
↓
double f(double x) {
        double r25575 = x;
        double r25576 = -0.034229541342750226;
        bool r25577 = r25575 <= r25576;
        double r25578 = 0.0357523883624483;
        bool r25579 = r25575 <= r25578;
        double r25580 = !r25579;
        bool r25581 = r25577 || r25580;
        double r25582 = 1.0;
        double r25583 = cos(r25575);
        double r25584 = r25582 - r25583;
        double r25585 = exp(r25584);
        double r25586 = log(r25585);
        double r25587 = sqrt(r25586);
        double r25588 = cbrt(r25584);
        double r25589 = fabs(r25588);
        double r25590 = r25587 * r25589;
        double r25591 = r25590 / r25575;
        double r25592 = 3.0;
        double r25593 = pow(r25582, r25592);
        double r25594 = pow(r25583, r25592);
        double r25595 = r25593 - r25594;
        double r25596 = r25583 + r25582;
        double r25597 = r25583 * r25596;
        double r25598 = r25582 * r25582;
        double r25599 = r25597 + r25598;
        double r25600 = r25595 / r25599;
        double r25601 = cbrt(r25600);
        double r25602 = sqrt(r25601);
        double r25603 = r25602 / r25575;
        double r25604 = r25591 * r25603;
        double r25605 = 0.001388888888888889;
        double r25606 = 4.0;
        double r25607 = pow(r25575, r25606);
        double r25608 = r25605 * r25607;
        double r25609 = 0.5;
        double r25610 = r25608 + r25609;
        double r25611 = 0.041666666666666664;
        double r25612 = 2.0;
        double r25613 = pow(r25575, r25612);
        double r25614 = r25611 * r25613;
        double r25615 = r25610 - r25614;
        double r25616 = r25581 ? r25604 : r25615;
        return r25616;
}

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 < -0.034229541342750226 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

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.4

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

Reproduce

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