Average Error: 39.7 → 16.6
Time: 7.0s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.05130406889125483 \cdot 10^{-26}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.496850468976921 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.05130406889125483 \cdot 10^{-26}:\\
\;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.496850468976921 \cdot 10^{-5}:\\
\;\;\;\;\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\

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

\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 <= -5.051304068891255e-26)) {
		VAR = ((double) (((double) log(((double) exp(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))))))) - ((double) cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 1.496850468976921e-05)) {
			VAR_1 = ((double) (((double) (0.16666666666666666 * ((double) (((double) pow(x, 3.0)) * eps)))) - ((double) (((double) (x * eps)) + ((double) (0.5 * ((double) pow(eps, 2.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) (((double) cos(x)) * ((double) cos(eps)))), 3.0)) - ((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)))) / ((double) (((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) * ((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) (((double) cos(x)) * ((double) cos(eps)))))))) + ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) * ((double) (((double) cos(x)) * ((double) cos(eps)))))))))) - ((double) cos(x))));
		}
		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 < -5.05130406889125483e-26

    1. Initial program 32.1

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

    3. Applied cos-sum4.0

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

    5. Applied add-log-exp4.1

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

    6. Applied add-log-exp4.3

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

    7. Applied diff-log4.3

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

    8. Simplified4.2

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

    if -5.05130406889125483e-26 < eps < 1.496850468976921e-5

    1. Initial program 48.8

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

    3. Applied cos-sum48.6

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

    5. Applied flip--48.6

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

    6. Taylor expanded around 0 32.2

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    if 1.496850468976921e-5 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum0.9

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

    5. Applied flip3--1.1

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

    6. Simplified1.1

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.05130406889125483 \cdot 10^{-26}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.496850468976921 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))