Average Error: 31.7 → 0.5
Time: 5.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03052087690052085813818827375598630169407:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}\right| \cdot \frac{\sqrt{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}}{x}\right)\\ \mathbf{elif}\;x \le 0.02841519458586902815988572967853542650118:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x \cdot x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03052087690052085813818827375598630169407:\\
\;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}\right| \cdot \frac{\sqrt{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}}{x}\right)\\

\mathbf{elif}\;x \le 0.02841519458586902815988572967853542650118:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x \cdot x}\\

\end{array}
double f(double x) {
        double r49541 = 1.0;
        double r49542 = x;
        double r49543 = cos(r49542);
        double r49544 = r49541 - r49543;
        double r49545 = r49542 * r49542;
        double r49546 = r49544 / r49545;
        return r49546;
}
↓
double f(double x) {
        double r49547 = x;
        double r49548 = -0.030520876900520858;
        bool r49549 = r49547 <= r49548;
        double r49550 = 1.0;
        double r49551 = cos(r49547);
        double r49552 = r49550 - r49551;
        double r49553 = sqrt(r49552);
        double r49554 = r49553 / r49547;
        double r49555 = 3.0;
        double r49556 = pow(r49552, r49555);
        double r49557 = cbrt(r49556);
        double r49558 = cbrt(r49557);
        double r49559 = fabs(r49558);
        double r49560 = exp(r49552);
        double r49561 = log(r49560);
        double r49562 = cbrt(r49561);
        double r49563 = sqrt(r49562);
        double r49564 = r49563 / r49547;
        double r49565 = r49559 * r49564;
        double r49566 = r49554 * r49565;
        double r49567 = 0.028415194585869028;
        bool r49568 = r49547 <= r49567;
        double r49569 = 0.001388888888888889;
        double r49570 = 4.0;
        double r49571 = pow(r49547, r49570);
        double r49572 = r49569 * r49571;
        double r49573 = 0.5;
        double r49574 = r49572 + r49573;
        double r49575 = 0.041666666666666664;
        double r49576 = 2.0;
        double r49577 = pow(r49547, r49576);
        double r49578 = r49575 * r49577;
        double r49579 = r49574 - r49578;
        double r49580 = pow(r49550, r49555);
        double r49581 = pow(r49551, r49555);
        double r49582 = r49580 - r49581;
        double r49583 = r49551 + r49550;
        double r49584 = r49551 * r49583;
        double r49585 = r49550 * r49550;
        double r49586 = r49584 + r49585;
        double r49587 = r49582 / r49586;
        double r49588 = r49547 * r49547;
        double r49589 = r49587 / r49588;
        double r49590 = r49568 ? r49579 : r49589;
        double r49591 = r49549 ? r49566 : r49590;
        return r49591;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.030520876900520858

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

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

    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. Simplified0.7

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

    12. Applied add-log-exp0.7

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

    13. Applied add-log-exp0.7

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

    14. Applied diff-log0.7

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

    15. Simplified0.7

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

    17. Applied add-cbrt-cube0.7

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

    18. Simplified0.7

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

    if -0.030520876900520858 < x < 0.028415194585869028

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.028415194585869028 < x

    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. Simplified1.2

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{x \cdot x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03052087690052085813818827375598630169407:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \left(\left|\sqrt[3]{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}\right| \cdot \frac{\sqrt{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}}{x}\right)\\ \mathbf{elif}\;x \le 0.02841519458586902815988572967853542650118:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))