Average Error: 39.3 → 0.5
Time: 7.4s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038847635134518256 \lor \neg \left(\varepsilon \leq 0.003604158899811683\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0038847635134518256 \lor \neg \left(\varepsilon \leq 0.003604158899811683\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)\\

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0038847635134518256) (not (<= eps 0.003604158899811683)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (*
    -2.0
    (*
     (sin (/ eps 2.0))
     (-
      (* (cos x) (+ (* eps 0.5) (* (pow eps 3.0) -0.020833333333333332)))
      (- (* 0.125 (* (sin x) (* eps 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 <= -0.0038847635134518256) || !(eps <= 0.003604158899811683)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = -2.0 * (sin(eps / 2.0) * ((cos(x) * ((eps * 0.5) + (pow(eps, 3.0) * -0.020833333333333332))) - ((0.125 * (sin(x) * (eps * 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 2 regimes

  2. if eps < -0.0038847635134518256 or 0.0036041588998116832 < eps

    1. Initial program 29.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.0038847635134518256 < eps < 0.0036041588998116832

    1. Initial program 49.3

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

    3. Applied diff-cos_binary64_22937.7

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

    4. Simplified0.6

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

    5. Taylor expanded around 0 0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\left(\sin x + 0.5 \cdot \left(\cos x \cdot \varepsilon\right)\right) - \left(0.125 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + 0.020833333333333332 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right)\right)\right)}\right)\]

    6. Simplified0.2

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos x \cdot \left(0.5 \cdot \varepsilon + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038847635134518256 \lor \neg \left(\varepsilon \leq 0.003604158899811683\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) - \left(0.125 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021025 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))