Average Error: 39.7 → 0.5
Time: 9.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00271818890310635:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029740925941223197:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(\varepsilon \cdot \sin x + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.00271818890310635:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\

\mathbf{elif}\;\varepsilon \leq 0.0029740925941223197:\\
\;\;\;\;\left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(\varepsilon \cdot \sin x + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00271818890310635)
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin eps) (sin x))))
   (if (<= eps 0.0029740925941223197)
     (-
      (+
       (* 0.041666666666666664 (* (cos x) (pow eps 4.0)))
       (* 0.16666666666666666 (* (sin x) (pow eps 3.0))))
      (+ (* eps (sin x)) (* 0.5 (* (cos x) (pow eps 2.0)))))
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00271818890310635) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(eps) * sin(x)));
	} else if (eps <= 0.0029740925941223197) {
		tmp = ((0.041666666666666664 * (cos(x) * pow(eps, 4.0))) + (0.16666666666666666 * (sin(x) * pow(eps, 3.0)))) - ((eps * sin(x)) + (0.5 * (cos(x) * pow(eps, 2.0))));
	} else {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	}
	return tmp;
}

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 < -0.00271818890310635

    1. Initial program 29.6

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

    3. Applied cos-sum_binary64_2120.8

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

    4. Applied associate--l-_binary64_160.8

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

    5. Simplified0.8

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

    if -0.00271818890310635 < eps < 0.00297409259412231971

    1. Initial program 50.1

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(\sin x \cdot \varepsilon + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)}\]

    if 0.00297409259412231971 < eps

    1. Initial program 29.4

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

    3. Applied cos-sum_binary64_2120.8

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

    4. Applied associate--l-_binary64_160.8

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

    5. Simplified0.8

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

    7. Applied associate--r+_binary64_140.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00271818890310635:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029740925941223197:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right) - \left(\varepsilon \cdot \sin x + 0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))