Average Error: 39.6 → 16.0
Time: 8.3s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2838162386962672 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.7353122816561398 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^{3}}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.2838162386962672 \cdot 10^{-8}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 3.7353122816561398 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^{3}}\\

\end{array}
double code(double x, double eps) {
	return ((double) (((double) cos(((double) (x + eps)))) - ((double) cos(x))));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -1.2838162386962672e-08)) {
		VAR = ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) log(((double) exp(((double) (((double) sin(x)) * ((double) sin(eps)))))))))) - ((double) cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 0.000373531228165614)) {
			VAR_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(eps, 4.0)))) - ((double) (((double) (x * eps)) + ((double) (0.5 * ((double) pow(eps, 2.0))))))));
		} else {
			VAR_1 = ((double) cbrt(((double) pow(((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) cos(x)))))), 3.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 3 regimes

  2. if eps < -1.2838162386962672e-08

    1. Initial program 30.0

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

    3. Applied cos-sum1.2

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

    5. Applied add-log-exp1.3

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

    if -1.2838162386962672e-08 < eps < 0.000373531228165614

    1. Initial program 49.2

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

    3. Applied cos-sum48.8

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

    5. Applied add-cbrt-cube48.8

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

    6. Simplified48.8

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

    7. Taylor expanded around 0 31.2

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

    if 0.000373531228165614 < eps

    1. Initial program 30.5

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

    3. Applied cos-sum0.8

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

    5. Applied add-cbrt-cube1.1

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

    6. Simplified1.1

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2838162386962672 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 3.7353122816561398 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))