Average Error: 39.6 → 16.1
Time: 7.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.02333196694931366 \cdot 10^{-13}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.16717710983658106 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.02333196694931366 \cdot 10^{-13}:\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.16717710983658106 \cdot 10^{-10}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -3.0233319669493137e-13)) {
		VAR = ((pow(((cos(x) * cos(eps)) - (sin(x) * sin(eps))), 3.0) - pow(cos(x), 3.0)) / ((((cos(eps) * cos(x)) - (sin(x) * sin(eps))) * ((pow(((cos(x) * cos(eps)) - (sin(x) * sin(eps))), 2.0) - pow(cos(x), 2.0)) / (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x)))) + (cos(x) * cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 1.167177109836581e-10)) {
			VAR_1 = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (eps * 0.5)));
		} else {
			VAR_1 = ((cos(x) * cos(eps)) - cbrt(pow(((sin(x) * sin(eps)) + 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 < -3.0233319669493137e-13

    1. Initial program 31.7

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

    3. Applied cos-sum1.8

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

    5. Applied flip3--2.0

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

    6. Simplified2.0

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

    8. Applied flip-+2.0

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

    9. Simplified2.0

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

    if -3.0233319669493137e-13 < eps < 1.167177109836581e-10

    1. Initial program 48.4

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

    2. Taylor expanded around 0 30.9

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

    3. Simplified30.9

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

    if 1.167177109836581e-10 < eps

    1. Initial program 30.7

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

    3. Applied cos-sum1.5

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

    4. Applied associate--l-1.5

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

    6. Applied add-cbrt-cube1.7

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

    7. Simplified1.7

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.02333196694931366 \cdot 10^{-13}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.16717710983658106 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))