Average Error: 39.6 → 0.5
Time: 8.9s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025355825843680958 \lor \neg \left(\varepsilon \leq 0.0034251863869845934\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} - 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0025355825843680958 \lor \neg \left(\varepsilon \leq 0.0034251863869845934\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0025355825843680958) (not (<= eps 0.0034251863869845934)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (+
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
    (*
     (cos x)
     (- (* 0.041666666666666664 (pow eps 4.0)) (* 0.5 (* eps eps)))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0025355825843680958) || !(eps <= 0.0034251863869845934)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) - (0.5 * (eps * eps))));
	}
	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 2 regimes

  2. if eps < -0.00253558258436809578 or 0.00342518638698459344 < eps

    1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.8

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

    5. Simplified0.8

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

    if -0.00253558258436809578 < eps < 0.00342518638698459344

    1. Initial program 48.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025355825843680958 \lor \neg \left(\varepsilon \leq 0.0034251863869845934\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} - 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))