Average Error: 39.6 → 16.5
Time: 7.9s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.53880641214754542 \cdot 10^{-21} \lor \neg \left(\varepsilon \le 1.16016725680204362 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \cos \varepsilon\right) - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon + \cos x\right) \cdot \left(\left(\sin x \cdot \sin \varepsilon + \cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.53880641214754542 \cdot 10^{-21} \lor \neg \left(\varepsilon \le 1.16016725680204362 \cdot 10^{-5}\right):\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \cos \varepsilon\right) - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon + \cos x\right) \cdot \left(\left(\sin x \cdot \sin \varepsilon + \cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\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 <= -9.538806412147545e-21) || !(eps <= 1.1601672568020436e-05))) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) cos(x)) * ((double) cos(eps)))), 2.0)) * ((double) (((double) cos(x)) * ((double) cos(eps)))))) - ((double) pow(((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) cos(x)))), 3.0)))) / ((double) (((double) (((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) cos(x)))) * ((double) (((double) (((double) (((double) sin(x)) * ((double) sin(eps)))) + ((double) cos(x)))) + ((double) (((double) cos(x)) * ((double) cos(eps)))))))) + ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) * ((double) (((double) cos(x)) * ((double) cos(eps))))))))));
	} else {
		VAR = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
	}
	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 2 regimes

  2. if eps < -9.53880641214754542e-21 or 1.16016725680204362e-5 < eps

    1. Initial program 31.2

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

    3. Applied cos-sum2.2

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

    4. Applied associate--l-2.3

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

    6. Applied flip3--2.4

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

    7. Simplified2.4

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

    9. Applied unpow32.4

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

    10. Simplified2.4

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

    if -9.53880641214754542e-21 < eps < 1.16016725680204362e-5

    1. Initial program 48.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Taylor expanded around 0 32.0

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

    3. Simplified32.0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.53880641214754542 \cdot 10^{-21} \lor \neg \left(\varepsilon \le 1.16016725680204362 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \cos \varepsilon\right) - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon + \cos x\right) \cdot \left(\left(\sin x \cdot \sin \varepsilon + \cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))