Average Error: 39.7 → 16.1
Time: 7.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.9118824918033931 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 2.2379838828549683 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.9118824918033931 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 2.2379838828549683 \cdot 10^{-7}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\

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

\end{array}
double f(double x, double eps) {
        double r58421 = x;
        double r58422 = eps;
        double r58423 = r58421 + r58422;
        double r58424 = cos(r58423);
        double r58425 = cos(r58421);
        double r58426 = r58424 - r58425;
        return r58426;
}
↓
double f(double x, double eps) {
        double r58427 = eps;
        double r58428 = -8.911882491803393e-16;
        bool r58429 = r58427 <= r58428;
        double r58430 = 2.2379838828549683e-07;
        bool r58431 = r58427 <= r58430;
        double r58432 = !r58431;
        bool r58433 = r58429 || r58432;
        double r58434 = x;
        double r58435 = cos(r58434);
        double r58436 = cos(r58427);
        double r58437 = r58435 * r58436;
        double r58438 = sin(r58434);
        double r58439 = sin(r58427);
        double r58440 = r58438 * r58439;
        double r58441 = 3.0;
        double r58442 = pow(r58440, r58441);
        double r58443 = cbrt(r58442);
        double r58444 = r58437 - r58443;
        double r58445 = r58444 - r58435;
        double r58446 = r58440 - r58435;
        double r58447 = r58437 - r58446;
        double r58448 = 0.041666666666666664;
        double r58449 = 4.0;
        double r58450 = pow(r58427, r58449);
        double r58451 = r58448 * r58450;
        double r58452 = r58434 * r58427;
        double r58453 = 0.5;
        double r58454 = 2.0;
        double r58455 = pow(r58427, r58454);
        double r58456 = r58453 * r58455;
        double r58457 = r58452 + r58456;
        double r58458 = r58451 - r58457;
        double r58459 = r58447 * r58458;
        double r58460 = r58459 / r58447;
        double r58461 = r58433 ? r58445 : r58460;
        return r58461;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -8.911882491803393e-16 or 2.2379838828549683e-07 < eps

    1. Initial program 31.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum1.7

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.8

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

    6. Applied add-cbrt-cube1.8

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

    7. Applied cbrt-unprod1.8

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

    8. Simplified1.8

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

    if -8.911882491803393e-16 < eps < 2.2379838828549683e-07

    1. Initial program 48.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum48.7

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied add-cbrt-cube48.7

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

    6. Applied add-cbrt-cube48.7

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

    7. Applied cbrt-unprod48.7

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

    8. Simplified48.7

      \[\leadsto \left(\cos x \cdot \cos \varepsilon - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}\right) - \cos x\]
    9. Using strategy rm

    10. Applied flip--48.7

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

    11. Simplified48.7

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

    12. Simplified48.7

      \[\leadsto \frac{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) \cdot \left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}{\color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)}}\]

    13. Taylor expanded around 0 31.2

      \[\leadsto \frac{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.9118824918033931 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 2.2379838828549683 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \cos x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))