Average Error: 39.7 → 0.8
Time: 6.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2118697881203442 \cdot 10^{-11} \lor \neg \left(\varepsilon \le 7.268025054873702 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin x \cdot \sin \varepsilon\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.2118697881203442 \cdot 10^{-11} \lor \neg \left(\varepsilon \le 7.268025054873702 \cdot 10^{-5}\right):\\
\;\;\;\;\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin x \cdot \sin \varepsilon\\

\end{array}
double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}
↓
double code(double x, double eps) {
	double temp;
	if (((eps <= -1.2118697881203442e-11) || !(eps <= 7.268025054873702e-05))) {
		temp = ((cos(x) * (cos(eps) - 1.0)) - (sin(x) * sin(eps)));
	} else {
		temp = ((-0.5 * pow(eps, 2.0)) - (sin(x) * sin(eps)));
	}
	return temp;
}

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 < -1.2118697881203442e-11 or 7.268025054873702e-05 < eps

    1. Initial program 30.3

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

    3. Applied cos-sum1.3

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

    4. Taylor expanded around inf 1.3

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

    6. Applied +-commutative1.3

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

    7. Applied associate--r+1.3

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

    9. Applied *-un-lft-identity1.3

      \[\leadsto \left(\cos \varepsilon \cdot \cos x - \color{blue}{1 \cdot \cos x}\right) - \sin x \cdot \sin \varepsilon\]

    10. Applied distribute-rgt-out--1.3

      \[\leadsto \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)} - \sin x \cdot \sin \varepsilon\]

    if -1.2118697881203442e-11 < eps < 7.268025054873702e-05

    1. Initial program 49.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Taylor expanded around inf 48.7

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

    6. Applied +-commutative48.7

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

    7. Applied associate--r+11.6

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

    8. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} - \sin x \cdot \sin \varepsilon\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2118697881203442 \cdot 10^{-11} \lor \neg \left(\varepsilon \le 7.268025054873702 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin x \cdot \sin \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))